###### Abstract

Let $S$ be a compact Hausdorff space and $X$ a complex manifold. We consider
the space $C(S,X)$ of continuous maps $S\to X$, and prove that any bounded
holomorphic function on this space can be continued to a holomorphic function,
possibly multivalued, on a larger space $B(S,X)$ of Borel maps. As an
application we prove two theorems about bounded holomorphic functions on
$C(S,X)$, one reminiscent of the Monodromy Theorem, the other of Liouville’s
Theorem.

## Introduction

One version of the Aron–Berner extension theorem considers a complex Banach
space $\mathfrak{X}$ and for every bounded holomorphic function on its unit
ball provides a holomorphic extension to the unit ball of the bidual (second
dual) $\mathfrak{X}^{\prime\prime}\supset\mathfrak{X}$. After the initial,
somewhat weaker result of Aron and Berner, the sharp form above was proved by
Davie and Gamelin. Aron and Berner also proved a similar theorem with the ball
in $\mathfrak{X}$ replaced by a general open subset of $\mathfrak{X}$, that
was sharpened subsequently by Zalduendo [AB78, DG89, Z90].—It is well
understood in infinite dimensional complex analysis that the result is tight
in many ways. While linear forms on $\mathfrak{X}$ can be extended to an
arbitrary superspace $\mathfrak{Y}$, this is no longer generally true for
holomorphic functions, not even for quadratic functions. Work of Dineen and
Josefson gives that extension even to $\mathfrak{X}^{\prime\prime}$ may fail
without the boundedness assumption. For example, a holomorphic function $f$ on
the space $\mathfrak{X}=c_{0}$ of complex sequences tending to $0$ extends to
a holomorphic function on $\mathfrak{X}^{\prime\prime}\approx l^{\infty}$ if
and only if $f$ is bounded on every bounded subset of $\mathfrak{X}$ [D71,
J78].

A natural question is whether similar extension is possible for complex Banach
manifolds $\mathfrak{X}$ rather than open subsets of Banach spaces. Of course,
before talking about extensions of holomorphic functions, one should first
construct a manifold $\mathfrak{X}^{\prime\prime}$ to play the role of the
bidual, apparently a hard problem. Dineen and Venkova addressed it when
$\mathfrak{X}$ is a Riemann domain over a certain type of Banach space, and
extended holomorphic functions to a Riemann domain lying above the bidual
[DV04]. But in this paper we do not attempt a corresponding construction of
$\mathfrak{X}^{\prime\prime}$ in general, and instead produce extensions in a
more specialized framework, of Banach manifolds that arise as mapping spaces.

Let us start with a finite dimensional connected complex manifold $X$ and a
compact Hausdorff space $S$. The space $C(S,X)$ of continuous maps
$\mathfrak{x}:S\to X$ has the natural structure of a complex Banach manifold,
see [L04, LS07] and section 2 here. (When working with $X$, we will use $x,y,$
etc. to denote points of $X$, and the corresponding fraktur fonts
$\mathfrak{x},\mathfrak{y},\dots$ to denote mappings into $X$.) Whatever the
bidual of $C(S,X)$ should be, it must have something to do with the bidual of
the Banach space $C(S)$ of continuous functions $S\to\mathbb{C}$. This bidual
has been described, in slightly different ways, first by Kakutani, then by
Gordon, as the space of continuous functions on a related space $T$ [G66,
K41]; the book [D+16] gives yet another variant of the construction of $T$ and
of the isomorphism $C(S)^{\prime\prime}\approx C(T)$. This suggests that the
bidual of a mapping space $\mathfrak{X}=C(S,X)$ should be
$\mathfrak{X}^{\prime\prime}=C(T,X)$. It turns out that the associations
$(S,X)\mapsto C(S,X)=\mathfrak{X}$ and $(S,X)\mapsto
C(T,X)=\mathfrak{X}^{\prime\prime}$ are parts of (bi)functors between which
there is a natural transformation that embeds $\mathfrak{X}$ as a complex
submanifold into $\mathfrak{X}^{\prime\prime}$. The question then would be
whether bounded holomorphic functions on $\mathfrak{X}$ extend to
$\mathfrak{X}^{\prime\prime}$.

This is still not the question we will answer in this paper, though. With an
application in mind, that we will discuss shortly, we rather consider
extension to a smaller manifold

(0.1) $B(S,X)=\\{\mathfrak{x}:S\to X\text{ is Borel}\mid\mathfrak{x}(S)\subset
X\text{ has compact closure}\\}.$

(A map $\mathfrak{x}:S\to X$ is Borel if preimages of Borel subsets of $X$ are
Borel.) The space of Borel maps in (0.1) is also a complex Banach manifold.
This is the space to which holomorphic functions on $C(S,X)\subset B(S,X)$ are
to be extended.

It turns out that unlike in the Aron–Berner extension theorem, instead of
holomorphic functions it is better to work with possibly multivalued
holomorphic functions, that is, holomorphic germs that admit analytic
continuations along paths. We will define the relevant notions in section 7\.
For the moment we first submit that the Aron–Berner extension provides for any
holomorphic function germ $\bf f$ on $C(S,X)$ at some $\mathfrak{x}_{1}$ a
holomorphic germ $\bf f^{\text{ab}}$ on $B(S,X)$ (see section 5). We will
denote the germ of a complex manifold $M$ at some $p\in M$ by $(M,p)$ and,
accordingly, a germ of a function on $M$ at $p\in M$ as ${\bf
f}:(M,p)\to\mathbb{C}$.

######  Theorem 0.1.

Let $\mathfrak{x}_{1}\in C(S,X)$. If a holomorphic germ ${\bf
f}:\big{(}C(S,X),\mathfrak{x}_{1}\big{)}\to\mathbb{C}$ can be analytically
continued along any path in $C(S,X)$ starting at $\mathfrak{x}_{1}$, and these
continuations are uniformly bounded, then ${\bf f}^{\text{ab}}$ can be
analytically continued along any path in $B(S,X)$ starting at
$\mathfrak{x}_{1}$.

Without the boundedness assumption the theorem would be false, already if
$X=\mathbb{C}$ and $S$ is either $[0,1]$, or it consists of a convergent
sequence and its limit. Interestingly, in the theorem even if the analytic
continuations of $\bf f$ define a single valued function on $C(S,X)$, the
continuations of $\bf f^{\text{ab}}$ may define a multi–valued function on
$B(S,X)$. We will discuss this and related examples in section 11.

Since $B(S,X)$ is connected, Theorem 0.1 has the paradoxical consequence that
if the germ $\bf f$ has uniformly bounded analytic continuations along paths
in $C(S,X)$ starting at $\mathfrak{x}_{1}$, then it has canonical
continuations to all of the possibly disconnected $C(S,X)$.

Spaces of continuous maps are, of course, central objects in topology, while
Borel maps occur rarely, if ever, and for good reason: The topology of
$B(S,X)$ is too directly related to the topology of $X$, see Lemma 10.2 (and
(7.1), Lemma 9.4a). But this is precisely the reason why the connection
between the two mapping spaces, described in Theorem 0.1, has implications
about the continuous mapping spaces themselves:

######  Theorem 0.2 (Monodromy Theorem).

Suppose $X$ is simply connected and $\mathfrak{x}_{1}\in C(S,X)$. Assume a
holomorphic germ ${\bf f}:\big{(}C(S,X),\mathfrak{x}_{1}\big{)}\to\mathbb{C}$
has analytic continuation along any path in $C(S,X)$ starting at
$\mathfrak{x}_{1}$. If these continuations are uniformly bounded, then
(a) they define a single valued holomorphic function on the component of
$\mathfrak{x}_{1}$ in $C(S,X)$.
(b) If, in addition, $X$ is compact, then this single valued function is in
fact constant.

We emphasize that $X$, and not as in the traditional Monodromy theorem the
source manifold $C(S,X)$, is assumed to be simply connected.—Both theorems
will be proved in greater generality in sections 9 and 10.

That holomorphic functions (bounded or not) on certain mapping spaces must be
constant for deeper reasons was first discovered by Dineen and Mellon [DM98].
Subsequently with Szabó we generalized to spaces of ‘based’ maps when $X$ is
rationally connected, e.g., a Grassmannian; if $S$ is a manifold, constancy
follows even for spaces of $C^{k}$ maps [L04, LS07]. Nonetheless, similarity
with Theorem 0.2b is formal only, and the underlying mechanisms are different.
That this must be so can be seen upon comparing what is covered here and
there. On the one hand, the manifolds $X$ of Theorem 0.2 are vastly more
general than the rationally connected manifolds of [LS07]; on the other, the
requirement of boundedness and that we work with spaces of continuous, rather
than $C^{k}$ maps, severely restricts the scope of Theorem 0.2b, say. But this
is inevitable. For general compact simply connected $X$ and a compact smooth
manifold $S$ the spaces $C^{k}(S,X)$ of $C^{k}$ maps may support nonconstant
holomorphic functions when $k\geq 1$. For example, a Borel measure $\mu$ on
the circle $S^{1}$ and a holomorphic section $\alpha$ of $(T^{*}X)^{\otimes
m}$, not identically vanishing on multivectors of form
$\xi\otimes\xi\otimes\dots\in(TX)^{\otimes m}$, induces a holomorphic function
$a:C^{1}(S^{1},X)\to\mathbb{C}$,

$a(\mathfrak{x})=\int_{S^{1}}\alpha(\dot{\mathfrak{x}}\otimes\dot{\mathfrak{x}}\otimes\cdots)\,d\mu,$

typically not even locally constant.

Related results have been proved by Aron, Galindo, Pinasco, and Zalduendo. In
[A+16] they consider, for a compact space $S$, holomorphic functions on $C(S)$
that are invariant under the group of linear maps $C(S)\to C(S)$ induced by
homeomorphisms $S\to S$. Among other things they find that when $S$ is a
circle or a closed disc, such functions must be constant. While their
assumptions are very different from ours: here boundedness, there invariance,
both proofs rely on the idea of Aron–Berner extension.

Contents. Section 1 discusses the notion of Borel vector bundles. These are
needed to endow the space $B(S,X)$ with the structure of a complex Banach
manifold. Sections 2 and 3 review and discuss basic properties of mapping
spaces and of the Aron–Berner extension in Banach spaces. Section 4 is about
extension in local models of mapping spaces: from spaces of continuous
sections of certain topological vector bundles to spaces of Borel sections. In
particular, we prove a naturality property of this extension, Corollary 4.6.
Section 5 then uses this naturality to extend holomorphic germs ${\bf f}$ on
$C(S,X)$ to holomorphic germs ${\bf f}^{\text{ab}}$ on $B(S,X)$. All of this
is rather straightforward. Sections 6 and 7 discuss notions of holomorphic
Banach bundles, flat norms, and analytic continuation. Theorem 7.5 generalizes
Theorem 0.1 to sections of certain Banach bundles. Section 8 features a key
result: The construction of a family of holomorphic maps into $X$ of balls in
finite dimensional vector spaces. These will be used in section 9 to reduce
analytic continuation in $B(S,X)$ to analytic continuation in Banach spaces.
The proofs of Theorems 0.1, 0.2, generalized, are given in sections 9 and 10.
Section 11 contains examples that show that the main theorems are optimal in
certain ways.

During the preparation of this paper I have profited from discussion with
Pablo Galindo.

Our basic reference to complex analysis in Banach spaces is Mujica’s book
[M86], and to infinite dimensional complex manifolds, [L98, Section 2]. The
infinite dimensional complex manifolds that we will encounter are what [L98]
calls rectifiable, and modeled on complex Banach spaces; all this means is
that they are locally biholomorphic to open subsets of Banach spaces. We
denote the set of holomorphic maps between complex manifolds $M,N$ by ${\cal
O}(M,N)$, and by ${\cal O}(M)$ when $N=\mathbb{C}$.

##  1 Borel vector bundles

While continuous mapping spaces $C(S,X)$ are Banach manifolds modeled on
spaces of continuous sections of topological vector bundles, the local models
of spaces $B(S,X)$ of Borel maps are spaces of sections of what we call Borel
vector bundles. In this section we develop the notion in the generality we
will need. Related but different notions, such as measurable fields and
bundles of Banach, especially Hilbert, spaces have been introduced before.

Let $S$ be a topological space and $B(S)$ the algebra of bounded Borel
functions $S\to\mathbb{C}$. Consider a map $\pi:V\to S$ of sets, with each
$V_{s}=\pi^{-1}s$, $s\in S$, endowed with the structure of a finite
dimensional complex vector space. The space of all sections of $\pi$, with
pointwise addition and multiplication by bounded Borel functions, forms a
$B(S)$–module. By a Borel field of (finite dimensional) vector spaces we mean
the datum of such $\pi:V\to S$ plus a $B(S)$–submodule $\Xi$ of the module of
all sections. As is customary, instead of $(\pi:V\to S,\Xi)$, often we just
write $\pi:V\to S$, or even just $V$, for a Borel field of vector spaces, and
$\Xi$ is only implied. If $\dim V_{s}$ is the same for all $s\in S$, we call
this number the rank of $V$.

Suppose $T\subset S$ is Borel. We write $V|T$ first for $\pi^{-1}T$, but also
for the Borel field $(\pi^{-1}T\to T,\Xi|T)$, where
$\Xi|T=\\{\xi|T:\xi\in\Xi\\}$. This is the restriction of $V$ to $T$.

A homomorphism of Borel fields of vector spaces $(V\to S,\Xi)$, $(W\to S,H)$
is a map $F:V\to W$, linear between $V_{s}$ and $W_{s}$, $s\in S$, such that
$F\circ\xi\in H$ for all $\xi\in\Xi$. A homomorphism is an isomorphism if it
has an inverse, which is also a homomorphism.

Any finite dimensional complex vector space $A$ defines a trivial Borel field
of vector spaces $S\times A\to S$ and $\Xi$ consisting of sections that
correspond to bounded Borel functions $S\to A$. Denote the space of linear
maps between vector spaces $A,B$ by $\text{Hom}\,(A,B)$. Any bounded Borel map
$f:S\to\text{Hom}\,(A,B)$ induces a homorphism

$F:S\times A\ni(s,a)\mapsto\big{(}s,f(s)a\big{)}\in S\times B$

of trivial Borel fields of vector spaces, and any homomorphism arises in this
way.

######  Definition 1.1.

A Borel vector bundle is a Borel field of vector spaces that is isomorphic to
a trivial Borel field of vector spaces.

Admittedly, this does not have the ring of “vector bundle”, that should be
trivial only “locally”. Still, if $S$ is the finite union of Borel sets
$S_{j}$, and for a Borel field of vector spaces $V\to S$ the restrictions
$V|S_{j}$ are isomorphic to trivial fields of the same rank, then $V$ itself
is isomorphic to a trivial Borel field of vector spaces.

Any topological vector bundle $\pi:V\to S$ over a compact base, with
equidimensional fibers, determines a Borel vector field $(\pi:V\to S,\Xi)$: a
section $\xi$ of $\pi$ is in $\Xi$ if in any topological trivialization over
some compact $K\subset S$, $V|K\approx K\times A$, it corresponds to a bounded
Borel function $K\to A$.—Further examples of Borel vector bundles can be
obtained by pullback. Suppose $(\pi:V\to S,\Xi)$ is a Borel vector bundle, and
$g:T\to S$ is a Borel map of topological spaces. Then $g^{*}V=\coprod_{t\in
T}V_{g(t)}$, with the obvious projection $\rho:g^{*}V\to T$, and the
$B(T)$–submodule $H$ generated by sections of form
$\eta(t)=\xi\big{(}g(t)\big{)}$, $\xi\in\Xi$, defines the pullback Borel
vector bundle $(g^{*}V\to T,H)$, also denoted $g^{*}V$. (Here we identify
fibers of the pullback of a bundle $V$ with the fibers of $V$ itself,
something we will do elsewhere in the paper as well.) Equivalently, $H$
consists of those sections $\eta$ of $\rho$ for which the map $T\ni
t\mapsto\eta(t)\in V_{g(t)}\subset V$ is bounded and Borel, these properties
measured using any trivialization $V\approx S\times A$.

A norm on a trivial Borel vector bundle $S\times A\to S$ is a Borel function
$|\,\,|:S\times A\to[0,\infty)$ whose restrictions to the fibers
$\\{s\\}\times A$ define norms $|\,\,|_{s}$ on $A$, and there is a
$c\in\mathbb{R}$ such that $|\,\,|_{s}\leq c|\,\,|_{t}$ for all $s,t\in S$.
Any isomorphism $F:S\times B\to S\times A$ pulls back the norm $|\,\,|$ to a
norm $|\,\,|\circ F$ on $S\times B\to S$. Therefore we can define the notion
of a norm on a general Borel vector bundle $V\to S$ as a function
$V\to[0,\infty)$ that, under some (or an arbitrary) isomorphism $V\approx
S\times A$, corresponds to a norm on $S\times A\to S$. Such a norm is
measurable with respect to the Borel $\sigma$–algebra of $V$, defined as the
pullback of the Borel $\sigma$–algebra of $S\times A$ by any isomorphism
$V\approx S\times A$.

Given a Borel vector bundle $(V\to S,\Xi)$, it will be convenient to denote
$\Xi$ by111That we use similar notation $B(S)$, $B(V)$—and later $C(S)$,
$C(V)$—for rather different objects should not confuse the reader. In the
former $S$ is a topological space and $B(S)$ the space bounded Borel functions
on it; $V$ is Borel vector bundle over $S$. Similarly, when $S$ is a compact
Hausdorff space and $V\to S$ a topological vector bundle, $C(S)$ is the space
of continuous functions on $S$, and $C(V)$ is the space of continuous
sections. $B(V)$, and refer to it as the space of bounded Borel sections of
$V$. A norm $|\,\,|$ on $V$ turns $B(V)$ into a Banach space, with norm
$||\xi||=\sup_{S}|\xi|$. The topology on $B(V)$ is independent of the choice
of $|\,\,|$.

We finish this section by relating analysis on a Borel vector bundle $\pi:V\to
S$ and on the Banach(able) space $B(V)$. Given $U\subset V$, a norm $|\,\,|=p$
on $V$, and $\varepsilon>0$, set

$U_{p,\varepsilon}=\\{u\in U:p(u)<1/\varepsilon\text{ and }U\text{ contains
all $v\in V_{\pi u}$ with }p(u-v)<\varepsilon\\}.$

If $q$ is another norm on $V$, then $U_{p,\varepsilon}\subset
U_{q,c\varepsilon}$ with some $c>0$ independent of $\varepsilon$. Define

(1.1) $\hat{U}=\text{int}\,\\{\xi\in B(V):\xi(S)\subset
U\\}=\bigcup_{\varepsilon>0}\\{\xi\in B(V):\xi(S)\subset
U_{p,\varepsilon}\\}.$

Consider another Borel vector bundle $W\to S$ and a fiberwise $H:U\to W$,
i.e., $H(U_{s})\subset W_{s}$, where $U_{s}=U\cap V_{s}$. Pick norms $p,q$ on
$V,W$.

######  Definition 1.2.

We say that $H$ is of bounded type if $q$ is bounded on $H(U_{p,\varepsilon})$
for all $\varepsilon>0$.

Clearly, bounded type are independent of the choice of $p,q$

######  Lemma 1.3.

Suppose the fibers $U_{s}\subset V_{s}$ are open. If $H:U\to W$ is Borel, of
bounded type, and $H|U_{s}$ are holomorphic for $s\in S$, then

$H^{\circ}:\hat{U}\ni\xi\mapsto H\circ\xi\in B(W)$

is holomorphic.

###### Proof.

That $H^{\circ}$ maps into $B(W)$ follows from the definition in a
straightforward way. To prove holomorphy, we can assume that $V=S\times A$,
$W=S\times B$ are trivial. We will write $d_{A}H$ for the differential of $H$
along $A$. Cauchy’s formula along one dimensional slices in $\\{s\\}\times A$
gives that partials of $H$ along $A$ are Borel and bounded on
$U_{p,\varepsilon}$. Using this for first derivatives gives that $H^{\circ}$
is continuous; for second derivatives, that the directional derivatives

$dH^{\circ}(\xi;\eta)=\lim_{\mathbb{R}\ni t\to
0}\big{(}H^{\circ}(\xi+t\eta)-H^{\circ}(\xi)\big{)}/t,$

exist for any $\xi\in\hat{U}$ and $\eta\in B(V)$, and
$dH^{\circ}(\xi;\eta)(s)$ can be computed as
$d_{A}H\big{(}\xi(s);\eta(s)\big{)}$. Now $S\times TA\to S$ and $S\times TB\to
S$ have a natural structure of a Borel vector bundle. Since the fiberwise
tangent bundle $T_{A}U=\coprod_{s\in S}T\big{(}U\cap(\\{s\\}\times A)\big{)}$
is a Borel subset of $S\times TA$, $dH_{A}:T_{A}U\to S\times TW$ is Borel, of
bounded type, and holomorphic along the fibers, $dH^{\circ}(\xi;\eta)$ is a
continuous function of $\xi,\eta$. It is also complex linear in $\eta$, which
proves that $H^{\circ}$ is holomorphic.

##  2 The mapping spaces

Henceforward we work with a fixed finite dimensional, connected complex
manifold $X$ and a compact Hausdorff space $S$. Write $TX\to X$ for the real
tangent bundle, on which the complex structure of $X$ induces the structure of
a holomorphic vector bundle. As in the Introduction, we let

$B(S,X)=\\{\mathfrak{x}:S\to X\text{ is Borel}\mid\mathfrak{x}(S)\subset
X\text{ has compact closure}\\},$

and endow it with the uniform topology. Basic neighborhoods of
$\mathfrak{x}\in B(S,X)$ are determined by open neighborhoods $D\subset
X\times X$ of the diagonal $\\{(x,x):x\in X\\}$, the corresponding open
neighborhood of $\mathfrak{x}$ being

(2.1) $D(\mathfrak{x})=\\{\mathfrak{y}\in
B(S,X):(\mathfrak{x}\times\mathfrak{y})(S)\text{ has compact closure in
}D\\}.$

Here
$(\mathfrak{x}\times\mathfrak{y})(s)=\big{(}\mathfrak{x}(s),\mathfrak{y}(s)\big{)}$.

This topology is metrizable. Take any metric $d$ on $X$ that induces its
topology; then the metric

(2.2)
$d_{S}(\mathfrak{x},\mathfrak{y})=\sup\\{d\big{(}\mathfrak{x}(s),\mathfrak{y}(s)\big{)}:s\in
S\\}$

induces the uniform topology of $B(S,X)$. (The $\sup$ in (2.2) is finite,
because $\mathfrak{x}(S),\mathfrak{y}(S)$ are contained in some compact
$K\subset X$ and $d$ is continuous on $K\times K$.) The topology restricts to
the compact–open topology on the subspace $C(S,X)$ of continuous maps.

The definition of a manifold structure on $B(S,X)$ involves the induced Borel
vector bundles

$\mathfrak{x}^{*}TX=\coprod_{s\in S}T_{\mathfrak{x}(s)}X\to
S,\qquad\mathfrak{x}\in B(S,X),$

see section 1, and their spaces $B(\mathfrak{x}^{*}TX)$ of bounded Borel
sections. The material in section 1 allows to treat topological vector bundles
only over compact bases as Borel vector bundles, and then pull them back by
Borel maps. Here in general we have to take any compact $K\subset X$
containing $\mathfrak{x}(S)$; then $TX|K$ determines a Borel vector bundle and
can be pulled back by $\mathfrak{x}$ to produce a Borel vector bundle, that we
denoted $\mathfrak{x}^{*}TX$.

If $D\subset X\times X$, $F$ is a function defined on $D$, and $x\in X$, we
let

(2.3) $D^{x}=\\{y\in X:(x,y)\in D\\}\quad\text{and}\quad F^{x}=F(x,\cdot).$

######  Lemma 2.1.

There are a neighborhood $D\subset X\times X$ of the diagonal and a
$C^{\infty}$ diffeomorphism $F$ between $D$ and a neighborhood of the zero
section in $TX$ with the following properties:
(a) $F^{x}$ maps $D^{x}$ biholomorphically on a convex open subset of
$T_{x}X$;
(b) $F^{x}(x)\in T_{x}X$ is the zero vector;
(c) Identifying the tangent spaces of the vector space $T_{x}X$ with $T_{x}X$
itself, the differential $F_{*}^{x}|_{x}:T_{x}D^{x}=T_{x}X\to T_{x}X$ is the
identity.

Moreover, if $X$ is relatively compact and open in another complex manifold
$M$, then $F$ can be chosen real analytic.

The inverse of $F$ is thus as good a substitute for a holomorphic exponential
map $TX\to X\times X$ as it gets. Denoting by pr${}_{2}:X\times X\to X$
projection on the second factor, accordingly we define

(2.4) $\exp=\text{pr}_{2}\circ F^{-1}:TX\to X,$

whose restriction to $T_{x}X$ is a biholomorphism $F^{x}(D^{x})\to D^{x}$,
inverse to $F^{x}$.

###### Proof.

The first part is [LS07, Lemma 1.1]. The second part is a special case of
[L04, Lemma 2.1], that (when $k=\omega$) takes a compact real analytic
manifold $V$, possibly with boundary, a real analytic map $\mathfrak{x}:V\to
M$, and relates neighborhoods of the graph of $\mathfrak{x}$ in $V\times M$
and of the zero section in the induced bundle $\mathfrak{x}^{*}TM$.—Note,
though, the different use of $X$ here and there.—To apply that result,
following Grauert [G58] embed our $M$ in some $\mathbb{R}^{N}$ as a closed
real analytic submanifold, choose $V\subset M$ a suitable sublevel set of the
Euclidean norm function, and $\mathfrak{x}$ the embedding of $V$ in $M$.

In fact, a minor modification of the proof in [L04] gives that $F$ can be
chosen real analytic for any $X$, whether $M$ as above exists or not.

We are now in the position to define holomorphic charts that turn $B(S,X)$
into a complex manifold. Given $D,F$ as in the lemma, a coordinate
neighborhood will be $D(\mathfrak{x})$ of (2.1). The map

(2.5) $\varphi_{\mathfrak{x}}:D(\mathfrak{x})\ni\mathfrak{z}\mapsto
F\circ(\mathfrak{x}\times\mathfrak{z})\in B(\mathfrak{x}^{*}TX)$

is a homeomorphism on a neighborhood of the zero section in the Banach space
$B(\mathfrak{x}^{*}TX)$. Since the inverse of $\varphi_{\mathfrak{x}}$ is
given by $\xi\mapsto\exp\circ\xi$, to show that the local coordinates (2.5)
are holomorphically related we need to check that for
$\mathfrak{x},\mathfrak{y}\in B(S,X)$ the map

(2.6) $\varphi_{\mathfrak{y}}\circ\varphi_{\mathfrak{x}}^{-1}:\xi\mapsto
F\circ\big{(}\mathfrak{y}\times(\exp\circ\xi)\big{)}$

is holomorphic wherever defined on $B(\mathfrak{x}^{*}TX)$. This is a special
case of the following result, that we will need later on as well.

######  Lemma 2.2.

Let $Y$ be another finite dimensional complex manifold, $u\subset S\times X$
and $U\subset S\times TX$ open,

$\psi:u\to Y\quad\text{and}\quad\Psi:U\to TY$

continuous. Assume that $\Psi$ holomorphically maps the fibers
$U\cap(\\{s\\}\times T_{x}X)$ to $T_{\psi(s,x)}Y$. Fix $\mathfrak{x}\in
B(S,X)$ such that the closure of $(\text{id}_{S}\times\mathfrak{x})(S)$ is in
$u$, and let $\mathfrak{U}\subset B(\mathfrak{x}^{*}TX)$ consist of those
sections $\xi$ for which the closure of $(\text{id}_{S}\times\xi)(S)$ is
contained in $U$. Then
$\mathfrak{y}=\psi\circ(\text{id}_{S}\times\mathfrak{x})\in B(S,Y)$; for
$\xi\in\mathfrak{U}$

(2.7) $\eta=\Psi\circ(\text{id}_{S}\times\xi)$

is a bounded Borel section of $\mathfrak{y}^{*}TY$; and the map

$\Psi^{\circ}:\mathfrak{U}\ni\xi\mapsto\Psi\circ(\text{id}_{S}\times\xi)\in
B(\mathfrak{y}^{*}TY)$

is holomorphic.

It is of course possible that $\mathfrak{U}$ is empty, in which case the last
statement is, well, vacuous.—The holomorphy of the map (2.6) follows from
Lemma 2.2 if we let $Y=X$, $\psi:S\times X\to X$ the projection, and
$\Psi(s,v)=F\big{(}\pi v,\exp(v)\big{)}$, independently of $s$.

###### Proof of Lemma 2.2.

The lemma can be reduced to Lemma 1.3, but it is easier to write out a direct
proof, quite similar to the proof of Lemma 1.3. That $\mathfrak{y},\eta$ are
Borel follows because the composition of Borel functions is Borel; they are
also bounded, because $\psi,\Psi$ are bounded on the relatively compact sets
$(\text{id}_{S}\times\mathfrak{x})(S)$, respectively,
$(\text{id}_{S}\times\xi)(S)$. Next, the map $\Psi^{\circ}$ is continuous at
any $\xi_{0}\in\mathfrak{U}$. This follows from the uniform continuity of
$\Psi$ over a compact neighborhood $K\subset U$ of the closure of
$(\text{id}_{S}\times\xi_{0})(S)$.

To prove $\Psi^{\circ}$ is complex differentiable at $\xi_{0}$, we use the
same $K$ and another compact neighborhood $L\subset\text{int}\,K$. Cauchy
estimates then give that partial derivatives of $\Psi$ along the fibers of
$TX$ are bounded on $L$. By Taylor’s formula this implies that, denoting by
$d_{2}\Psi$ the differential of $\Psi$ along the manifolds $\\{s\\}\times
T_{x}X$, the derivative of $\Psi^{\circ}$ at $\xi_{0}$ in the direction
$\xi\in B(\mathfrak{x}^{*}TX)$

$d\Psi^{\circ}(\xi_{0};\xi)=\lim_{\mathbb{C}\ni t\to
0}\big{(}\Psi^{\circ}(\xi_{0}+t\xi)-\Psi^{\circ}(\xi_{0})\big{)}/t$

exists, and can be computed as

$d\Psi^{\circ}(\xi_{0};\xi)(s)=d_{2}\Psi\big{(}s,\xi_{0}(s);\xi(s)\big{)},$

cf. (2.7). One now argues, as with $\Psi^{\circ}$ itself, that
$d\Psi^{\circ}(\xi_{0};\cdot):B(\mathfrak{x}^{*}TX)\to B(\mathfrak{y}^{*}TY)$
is continuous. Hence $\Psi^{\circ}$ is complex differentiable at $\xi_{0}$,
and therefore holomorphic on $\mathfrak{U}$.

As we saw, one consequence of Lemma 2.2 is that the local coordinates
$\varphi_{\mathfrak{x}}$, $\mathfrak{x}\in B(S,X)$, are holomorphically
related, and endow $B(S,X)$ with the structure of a complex manifold, modeled
on Banach spaces of form $B(\mathfrak{x}^{*}TX)$.

For example, if $S=\\{s_{1},\dots,s_{m}\\}$ is finite, then

$B(S,X)\ni\mathfrak{x}\mapsto\big{(}\mathfrak{x}(s_{1}),\dots,\mathfrak{x}(s_{m})\big{)}\in
X^{m}$

is a biholomorphism. In general, the correspondence $(S,X)\mapsto B(S,X)$ is
natural: a continuous map $a:T\to S$ and a holomorphic map $X\to Y$ induce, by
composition, a holomorphic map $B(S,X)\to B(T,Y)$. More generally instead of a
holomorphic map we can use a family of holomorphic maps $\Phi:S\times X\to Y$.
We formulate the result only when $a:T=S\to S$ is the identity.

######  Lemma 2.3.

Let $\Phi:S\times X\to Y$ be continuous. If for each $s\in S$ the map
$\Phi(s,\cdot)$ is holomorphic, then

$\Phi^{\circ}:B(S,X)\ni\mathfrak{x}\mapsto\Phi\circ(\text{id}_{S}\times\mathfrak{x})\in
B(S,Y)$

is holomorphic.

###### Proof.

Let $D^{\prime}\subset Y\times Y$ and $F^{\prime}:D^{\prime}\to TY$ be as in
Lemma 2.1, but with $Y$ rather than $X$. If $\varphi_{\mathfrak{x}}$ is a
local coordinate on $B(S,X)$ as in (2.5),
$\mathfrak{y}=\Phi^{\circ}(\mathfrak{x})$, and
$\varphi^{\prime}_{\mathfrak{y}}$ is a local coordinate on $B(S,Y)$ at
$\mathfrak{y}$, we need to to check that the map

$\varphi^{\prime}_{\mathfrak{y}}\circ\Phi^{\circ}\circ\varphi_{\mathfrak{x}}^{-1}:\xi\mapsto
F^{\prime}\circ\big{(}\mathfrak{y}\times\Phi\circ(\text{id}_{S}\times\exp\circ\xi)\big{)}\in
B(\mathfrak{y}^{*}TY)$

is holomorphic where defined on $B(\mathfrak{x}^{*}TX)$. But this follows from
Lemma 2.2 if we choose $\Psi(s,v)=F^{\prime}\big{(}\pi v,\Phi(s,\exp
v)\big{)}$.

The complex structure on $B(S,X)$ was defined in terms of $D\subset X\times X$
and $F:D\to TX$. Since Lemma 2.3 applies with any choice of $D,F$ (and
$D^{\prime},F^{\prime}$), it follows by taking $Y=X$ and $\Phi(s,x)=x$ that,
in the end, different choices of $D,F$ give rise to the same complex structure
on $B(S,X)$.

Continuous maps $S\to X$ form a complex submanifold $C(S,X)\subset B(S,X)$,
modeled on Banach spaces $C(\mathfrak{x}^{*}TX)$ of continuous sections,
$\mathfrak{x}\in C(S,X)$. It is closed as a subset of $B(S,X)$, but not a
direct submanifold: while locally the pair $C(S,X)\subset B(S,X)$ is
biholomorphic to the pair $C(\mathfrak{x}^{*}TX)\subset
B(\mathfrak{x}^{*}TX)$, the space $C(\mathfrak{x}^{*}TX)$ will not in general
have a closed complement in $B(\mathfrak{x}^{*}TX)$. It is this circumstance
that makes even the local holomorphic extension problem from $C(S,X)$ to
$B(S,X)$ difficult (but solved by the Aron–Berner extension theorem).

An $\mathfrak{x}_{0}\in C(S,X)$ and a closed $A\subset S$ determine spaces of
based maps

$\mathfrak{B}=B(S,X,A,\mathfrak{x}_{0})\subset
B(S,X)\quad\text{and}\quad\mathfrak{C}=C(S,X,A,\mathfrak{x}_{0})\subset
C(S,X)\cap\mathfrak{B}.$

$\mathfrak{B}=\\{\mathfrak{x}\in B(S,X):\mathfrak{x}|A=\mathfrak{x}_{0}|A\\}$,
and $\mathfrak{C}$ denotes the connected component of $\mathfrak{x}_{0}$ in
$\\{\mathfrak{x}\in C(S,X):\mathfrak{x}|A=\mathfrak{x}_{0}|A\\}$. These are
closed complex submanifolds of $B(S,X)$, resp. $C(S,X)$, modeled on subspaces

$B_{A}(\mathfrak{x}^{*}TX)\subset
B(\mathfrak{x}^{*}TX),\quad\text{respectively}\quad
C_{A}(\mathfrak{x}^{*}TX)\subset C(\mathfrak{x}^{*}TX),$

of sections that vanish on $A$. It is the manifolds $\mathfrak{B}$ and
$\mathfrak{C}\subset\mathfrak{B}$ that are the targets of our investigations.

##  3 Aron–Berner extension

Let $\mathfrak{X},\mathfrak{Y}$ be Banach spaces, $x_{0}\in\mathfrak{X}$,
$r>0$, and

$B=\\{x\in\mathfrak{X}:||x-x_{0}||<r\\},\qquad
B^{\prime\prime}=\\{\xi\in\mathfrak{X}^{\prime\prime}:||\xi-x_{0}||<r\\}.$

In this section we review how to extend a bounded $f\in{\cal
O}(B,\mathfrak{Y})$ to a bounded function $f^{\text{AB}}\in{\cal
O}(B^{\prime\prime},\mathfrak{Y}^{\prime\prime})$, with the same bound.

The procedure starts with a(n always continuous) $k$–linear form
$T:\mathfrak{X}^{\oplus k}\to\mathbb{C}$, $k=0,1,\dots$, and constructs its
$k$–linear Aron–Berner extension $\bar{T}:\mathfrak{X}^{\prime\prime\oplus
k}\to\mathbb{C}$; the association $T\mapsto\bar{T}$ is linear. When $k=0$, $T$
is constant, and the extension is the same constant; when $k=1$,
$\bar{T}:\mathfrak{X}^{\prime\prime}\to\mathbb{C}$ is the second transpose of
$T$. In general, to define $\bar{T}$ by recurrence, suppose $\bar{T}$ exists
for $k$–linear operators $T$; then take a $(k+1)$–linear $T$. For any choice
of $x_{1},\dots,x_{k}\in\mathfrak{X}$ the second transpose of
$T(x_{1},\dots,x_{k},\cdot):\mathfrak{X}\to\mathbb{C}$ is a linear form
$T^{\prime\prime}(x_{1},\dots,x_{k},\cdot):\mathfrak{X}^{\prime\prime}\to\mathbb{C}$.
For each $\xi\in\mathfrak{X}^{\prime\prime}$

$T_{\xi}=T^{\prime\prime}(\cdot,\dots,\cdot,\xi):\mathfrak{X}^{\oplus
k}\to\mathbb{C}$

is $k$–linear, and depends linearly on $\xi$. The extensions
$\bar{T}_{\xi}:\mathfrak{X}^{\prime\prime\oplus k}\to\mathbb{C}$ then define

(3.1)
$\bar{T}(\xi_{1},\dots,\xi_{k+1})=\bar{T}_{\xi_{k+1}}(\xi_{1},\dots,\xi_{k}),\qquad\xi_{j}\in\mathfrak{X}^{\prime\prime}.$

More generally, if $T:\mathfrak{X}^{\oplus k}\to\mathfrak{Y}$ is a $k$–linear
operator, for each $l\in\mathfrak{Y}^{\prime}$ we consider the $k$–linear form
$T_{l}=lT$. The extensions constructed above, for each
$\xi_{j}\in\mathfrak{X}^{\prime\prime}$ produce a linear form
$l\mapsto\bar{T}_{l}(\xi_{1},\dots,\xi_{k})$ on $\mathfrak{Y}^{\prime}$, i.e.,
an element $\bar{T}(\xi_{1},\dots,\xi_{k})\in\mathfrak{Y}^{\prime\prime}$.
This defines the $k$–linear extension
$\bar{T}:\mathfrak{X}^{\prime\prime\oplus k}\to\mathfrak{Y}^{\prime\prime}$ of
$T$. The standard cautionary remark here is that if the arguments of $T$ are
permuted to produce a $k$–linear operator $T_{1}$, then $\bar{T}_{1}$ is not
in general the same as $\bar{T}$ with arguments correspondingly permuted.

The extension
$\bar{T}:\mathfrak{X}^{\prime\prime}\to\mathfrak{Y}^{\prime\prime}$ of a
linear operator $T$ is continuous when both
$\mathfrak{X}^{\prime\prime},\mathfrak{Y}^{\prime\prime}$ are endowed with the
weak* topology. The extension of multilinear operators is continuous only in a
weaker sense. Let us say that a function $F:\mathfrak{X}^{\prime\prime\oplus
k}\to\mathfrak{Y}^{\prime\prime}$ is slightly continuous to mean that for any
$j=1,\dots,k$, if $\xi_{1},\dots,\xi_{j-1}\in\mathfrak{X}$ and
$\xi_{j+1},\dots,\xi_{k}\in\mathfrak{X}^{\prime\prime}$ are fixed, then
$F(\xi_{1},\dots,\xi_{j},\dots,\xi_{k})\in\mathfrak{Y}^{\prime\prime}$ depends
continuously on $\xi_{j}\in\mathfrak{X}^{\prime\prime}$ in the weak* topology
of $\mathfrak{X}^{\prime\prime},\mathfrak{Y}^{\prime\prime}$. The following is
not new:

######  Lemma 3.1.

The extension $\bar{T}:\mathfrak{X}^{\prime\prime\oplus
k}\to\mathfrak{Y}^{\prime\prime}$ of a $k$–linear $T:\mathfrak{X}^{\oplus
k}\to\mathfrak{Y}$ is slightly continuous.

###### Proof.

It suffices to prove when $\mathfrak{Y}=\mathbb{C}$, that can be done by
induction. When $k=0$, $\bar{T}$ is constant. When $k=1$, $\bar{T}$ is the
second transpose of $T$, and the claimed continuity follows directly from the
definitions. Suppose the lemma holds for a certain $k\geq 1$ and let $T$ be
$(k+1)$–linear. To show that $\bar{T}$ is slightly continuous, we have to
check a certain continuity property for all $j=1,\dots,k+1$. When $j\leq k$,
this property follows from the induction hypothesis as $\bar{T}_{\xi_{k+1}}$
is slightly continuous, cf. (3.1). If $j=k+1$, and
$\xi_{i}=x_{i}\in\mathfrak{X}$, $i\leq k$, then again
$\bar{T}(x_{1},\dots,x_{k},\xi_{k+1})=T_{\xi_{k+1}}(x_{1},\dots,x_{k})=T^{\prime\prime}(x_{1},\dots,x_{k},\xi_{k+1})$
depends continuously on $\xi_{k+1}\in\mathfrak{X}^{\prime\prime}$.

Since $\mathfrak{X}$ is weak* dense in $\mathfrak{X}^{\prime\prime}$
(Goldstine’s theorem, [DS88, p. 424]), it follows that in the weak* topologies
of $\mathfrak{X}^{\prime\prime},\mathfrak{Y}^{\prime\prime}$

(3.2)
$\bar{T}(\xi_{1},\dots,\xi_{k})=\lim_{x_{1}\to\xi_{1}}\lim_{x_{2}\to\xi_{2}}\dots\lim_{x_{k}\to\xi_{k}}T(x_{1},\dots,x_{k}),\qquad\xi_{1},\dots,\xi_{k}\in\mathfrak{X}^{\prime\prime},$

which is often taken as the definition of $\bar{T}$, e.g., in [AB78].

We return to the construction of the Aron–Berner extension of $f$. The
construction is compatible with translations; for this reason we assume that
the center of the balls $B,B^{\prime\prime}$ is $x_{0}=0$. Any homogeneous
polynomial $P:\mathfrak{X}\to\mathfrak{Y}$ can be represented as
$P(x)=T(x,\dots,x)$ with a unique symmetric multilinear
$T:\mathfrak{X}^{\oplus k}\to\mathfrak{Y}$; we define
$P^{\text{AB}}(\xi)=\bar{T}(\xi,\dots,\xi)\in\mathfrak{Y}^{\prime\prime}$, a
homogeneous polynomial on $\mathfrak{X}^{\prime\prime}$. The correspondence
$P\mapsto P^{\text{AB}}$ can be extended by linearity to the space of all
polynomials $P:\mathfrak{X}\to\mathfrak{Y}$. We need the fact that, using
$\|\,\,\|$ for norm on $\mathfrak{Y}$ and on its iterated dual spaces,

(3.3) $\sup_{B^{\prime\prime}}||P^{\text{AB}}||=\sup_{B}||P||.$

This was proved by Davie and Gamelin [DG89] when $\mathfrak{Y}=\mathbb{C}$.
Carando in [C99] observed that (3.3) holds for homogeneous polynomials; that
it is true in general easily follows from [DG89]. Indeed, assume that
$\sup_{B}||P||=1$. If $l\in\mathfrak{Y}^{\prime}$ has norm $\leq 1$, then
$P_{l}=lP:\mathfrak{X}\to\mathbb{C}$ is a polynomial, $\sup_{B}|P_{l}|\leq
1,\text{ and }P_{l}^{\text{AB}}=lP^{\text{AB}}.$ In the latter formula $l$
acts on $\mathfrak{Y}^{\prime\prime}$ by transposition. By [DG89], then

(3.4) $\sup_{B^{\prime\prime}}|lP^{\text{AB}}|\leq 1\quad\text{for}\quad
l\in\mathfrak{Y}^{\prime},\,\,||l||\leq 1.$

But by Goldstine’s theorem [DS88, p. 424] the unit ball of
$\mathfrak{Y}^{\prime}$ is weak* dense in the unit ball of
$\mathfrak{Y}^{\prime\prime\prime}$, so that (3.4) holds for
$l\in\mathfrak{Y}^{\prime\prime\prime}$. Hence
$\sup_{B^{\prime\prime}}||P^{\text{AB}}||\leq 1$. The opposite inequality
being obvious, (3.3) holds.

The following result, a slight improvement on [AB78, DG89], is known in the
field:

######  Theorem 3.2.

Let $f\in{\cal O}(B,\mathfrak{Y})$ be bounded, $f=\sum_{k=0}^{\infty}P_{k}$
its expansion in homogeneous polynomials, $\deg P_{k}=k$. Then the series

$f^{\text{AB}}=\sum_{k=0}^{\infty}P_{k}^{\text{AB}}$

converges uniformly on concentric balls $B_{\rho}^{\prime\prime}\subset
B^{\prime\prime}$ of radius $\rho<r$; it represents the Aron–Berner extension
$f^{\text{AB}}\in{\cal O}(B^{\prime\prime},\mathfrak{Y}^{\prime\prime})$ of
$f$. In addition,

(3.5) $\sup_{B^{\prime\prime}}||f^{\text{AB}}||=\sup_{B}||f||.$

###### Proof.

$P_{k}$ is given by

$P_{k}(x)=\frac{1}{2\pi}\int_{0}^{2\pi}f(e^{it}x)e^{-ikt}\,dt.$

This and (3.3) imply

$\sup_{B^{\prime\prime}}||P_{k}^{\text{AB}}||=\sup_{B}||P_{k}||\leq\sup_{B}||f||.$

This in turn implies, in view of the homogeneity of $P_{k}^{\text{AB}}$, that
$\sum_{k}P_{k}^{\text{AB}}$ converges uniformly on
$B^{\prime\prime}_{\rho}\subset B^{\prime\prime}$, $\rho<r$, hence represents
a holomorphic function
$f^{\text{AB}}:B^{\prime\prime}\to\mathfrak{Y}^{\prime\prime}$. The estimate
(3.3), with $P$ a partial sum, yields

$\sup_{B^{\prime\prime}_{\rho}}\Big{\|}\sum_{k=0}^{m}P_{k}^{\text{AB}}\Big{\|}\leq\sup_{B}\Big{\|}\sum_{k=0}^{m}P_{k}\Big{\|}.$

Letting first $m\to\infty$ then $\rho\to r$, (3.5) follows.

Of course, $f^{\text{AB}}$ is not the unique holomorphic extension of $f$ to
$B^{\prime\prime}$. But it is natural in a sense. To formulate a theorem, we
need the notion of (Arens) regularity:

######  Definition 3.3.

A Banach space $\mathfrak{Y}$ is regular if for any Banach space
$\mathfrak{Z}$ (or, equivalently, only for $\mathfrak{Z}=\mathbb{C}$), the
extensions of any bilinear operator
$T:\mathfrak{Y}\oplus\mathfrak{Y}\to\mathfrak{Z}$ and of $T_{1}(x,y)=T(y,x)$
satisfy $\bar{T}_{1}(\xi,\eta)=\bar{T}(\eta,\xi)$ for all
$\xi,\eta\in\mathfrak{Y}$.

Induction then shows that for any $k$–linear $T:\mathfrak{Y}^{\oplus
k}\to\mathfrak{Z}$ and any permutation $\sigma$ of $\\{1,\dots,k\\}$, the
operator $T_{1}(x_{1},\dots,x_{k})=T(x_{\sigma(1)},\dots,x_{\sigma(k)})$
satisfies
$\bar{T}_{1}(\xi_{1},\dots,\xi_{k})=\bar{T}(\xi_{\sigma(1)},\dots,\xi_{\sigma(k)})$.
Since $\bar{T}(\cdot,\xi_{2},\dots,\xi_{k})$ is always continuous in the weak*
topologies, in regular spaces $\bar{T}$ is separately continuous in all its
variables. Ülger shows in [U87] that any C* algebra is regular; later on we
will need a minor generalization of this.

######  Theorem 3.4.

In addition to assumptions and notation above, let $y_{0}\in\mathfrak{Y}$,
$s>0$, and $B_{\mathfrak{Y}}=\\{y\in\mathfrak{Y}:||y-y_{0}||<s\\}.$ Suppose
$\mathfrak{Z}$ is a third Banach space, $f:B\to B_{\mathfrak{Y}}$ is
holomorphic, $f(x_{0})=y_{0}$, and $g:B_{\mathfrak{Y}}\to\mathfrak{Z}$ is
bounded and holomorphic. Then $(g\circ f)^{\text{AB}}=g^{\text{AB}}\circ
f^{\text{AB}}$, provided $g$ is a first degree polynomial, or $\mathfrak{Y}$
is regular.

###### Proof.

We will prove under the assumption that $\mathfrak{Y}$ is regular. The other
case follows by similar, but much simplified calculations. By translations we
can arrange that $x_{0}=0$ and $f(x_{0})=y_{0}=0$. First we assume that $f$
and $g$ are polynomials, $g$ homogeneous of degree $k$. This means that there
are symmetric multilinear operators $T:\mathfrak{Y}^{\oplus k}\to\mathfrak{Z}$
and $U_{i}:\mathfrak{X}^{\oplus i}\to\mathfrak{Y}$, $i=0,1,\dots,h$, such that

$\displaystyle g(y)=T(y,\dots,y),\qquad
f(x)=\sum_{i=0}^{h}U_{i}(x,\dots,x),\qquad\text{and therefore}$ (3.6)
$\displaystyle(g\circ f)(x)=T\big{(}f(x),\dots,f(x)\big{)}=\sum
T\big{(}U_{j_{1}}(x,\dots,x),U_{j_{2}}(x,\dots,x),\dots\big{)},$

summation over $j_{1},\dots,j_{k}\leq h$. The function (3.6) is a polynomial
in $x$. Its $j$’th degree component is obtained from the symmetrization of the
$j$–linear operator

$S_{j}(x_{1},\dots,x_{j})=\sum_{j_{1}+\dots+j_{k}=j}T\big{(}U_{j_{1}}(x_{1},\dots,x_{j_{1}}),U_{j_{2}}(x_{j_{1}+1},\dots,x_{j_{1}+j_{2}}),\dots\big{)}$

by setting $x=x_{1}=\dots=x_{j}$. The symmetrization
$S_{j,\text{sym}}(x_{1},\dots,x_{j})$ is

(3.7)
$\dfrac{1}{j!}\sum_{\sigma}\sum_{j_{1}+\dots+j_{k}=j}T\big{(}U_{j_{1}}(x_{\sigma(1)},\dots,x_{\sigma(j_{1})}),U_{j_{2}}(x_{\sigma(j_{1}+1)},\dots,x_{\sigma(j_{1}+j_{2})}),\dots\big{)},$

the first sum over all permutations $\sigma$ of $\\{1,\dots,j\\}$. Now

(3.8) $\begin{gathered}g^{\text{AB}}(\eta)=\bar{T}(\eta,\dots,\eta),\qquad
f^{\text{AB}}(\xi)=\sum_{i}\bar{U}_{i}(\xi,\dots,\xi),\qquad\text{and by
(3.2)}\\\ (g\circ
f)^{\text{AB}}(\xi)=\sum_{j}\lim_{x_{1}\to\xi}\lim_{x_{2}\to\xi}\dots\lim_{x_{j}\to\xi}S_{j,\text{sym}}(x_{1},\dots,x_{j}).\end{gathered}$

We compute this iterated limit for the generic term in (3.7). Since each
$U_{i}$ is symmetric, we may restrict attention to $\sigma$ such that
$\sigma(1)<\sigma(2)<\dots<\sigma(j_{1})$,
$\sigma(j_{1}+1)<\dots<\sigma(j_{1}+j_{2})$, and so on (shuffles). Then the
arguments of $T$ in this generic term tend to
$\bar{U}_{j_{1}}(\xi,\dots,\xi)$, $\bar{U}_{j_{2}}(\xi,\dots,\xi)$, etc., and
since $T$ is separately continuous, the limit of the generic term is
$\bar{T}\big{(}\bar{U}_{j_{1}}(\xi,\dots,\xi),\bar{U}_{j_{2}}(\xi,\dots\xi),\dots\big{)}$.
In light of (3.8) and (3.7) therefore the limit in (3.8) is indeed
$\bar{T}\big{(}\sum_{0}^{h}\bar{U}_{i}(\xi,\dots,\xi),\dots,\sum_{0}^{h}\bar{U}(\xi,\dots,\xi)\big{)}$.

Next we let $f$ be arbitrary, but keep $g$ $k$–homogeneous. We expand $f$ in a
homogeneous series. The partial sums $f_{j}$ of the series, polynomials, tend
to $f$ as $j\to\infty$, uniformly on concentric balls $\subset B$ of radius
$<r$. By (3.5) and by what we have proved,

$(g\circ f)^{\text{AB}}=\lim_{j\to\infty}(g\circ
f_{j})^{\text{AB}}=\lim_{j\to\infty}g^{\text{AB}}\circ
f_{j}^{\text{AB}}=g^{\text{AB}}\circ f^{\text{AB}}.$

Finally, a general bounded holomorphic $g:B_{\mathfrak{Y}}\to\mathfrak{Z}$ can
be expanded in a homogeneous series $g=\sum_{k}g_{k}$, converging uniformly on
concentric balls $\subset B_{\mathfrak{Y}}$ of radius $<s$. Again using (3.5),
on some neighborhood of $0\in B^{\prime\prime}$

$(g\circ f)^{\text{AB}}=\lim_{k\to\infty}\big{(}\sum_{j=0}^{k}g_{j}\circ
f)^{\text{AB}}=\lim_{k\to\infty}\sum_{j=0}^{k}g_{j}^{\text{AB}}\circ
f^{\text{AB}}=g^{\text{AB}}\circ f^{\text{AB}},$

and $(g\circ f)^{\text{AB}}=g^{\text{AB}}\circ f^{\text{AB}}$ follows by
analytic continuation.

##  4 Extensions from certain $C(S)$–modules

In this section we restrict ourselves to certain Banach spaces $\mathfrak{X}$
that are modules over $C(S)$. These modules arise from a topological complex
vector bundle $V\to S$, of finite rank, over the compact space $S$; they are
the space $C(V)$ of continuous sections of $V$ and certain subspaces. We embed
the space $B(V)$ of bounded Borel sections into the bidual of $C(V)$, and
study the restriction $f^{\text{ab}}$ to $B(V)$ (and subspaces) of the
Aron–Berner extension $f^{\text{AB}}$ from $C(V)$ (and subspaces). The
extension $f\mapsto f^{\text{ab}}$ has a certain naturality property. Since
the mapping spaces $\mathfrak{B},\mathfrak{C}$ are modeled on modules studied
here, the results of this section directly apply to provide local
Aron–Berner–type extensions in those mapping spaces.

By a norm on $V$ we mean a continuous function $|\,\,|:V\to[0,\infty)$ that
restricts to a vector space norm on each fiber $V_{s}$, $s\in S$. Given a
closed $A\subset S$ and a norm on $V$, we consider the Banach spaces

$B_{A}(V)=\\{\text{bounded Borel sections $\xi$ of
}V:\xi|A=0\\},\qquad||\xi||=\sup_{S}|\xi|,$

and $C_{A}(V)\subset B_{A}(V)$ the subspace of continuous sections.
Multiplication by functions turns these spaces into modules over the Banach
algebra $C(S)$. Thus $C_{\emptyset}(V)=C(V)$, $B_{\emptyset}(V)=B(V)$.

The main point will be the construction of a canonical embedding $B_{A}(V)\to
C_{A}(V)^{\prime\prime}$. We start by describing the dual $C_{A}(V)^{\prime}$
in terms of measures. Let $M(S\setminus A)$ denote the space of complex valued
regular Borel measures on $S\setminus A$. This is also a $C(S)$–module: the
product $fm$ of $f\in C(S)$ with a measure $m$ is given by

$(fm)(T)=\int_{T}f\,dm,\qquad T\subset S\setminus A\text{ Borel};$

thus $d(fm)/dm=f$. Let $C_{0}(S\setminus A)$ be the space of continuous
complex functions on the locally compact space $S\setminus A$ that vanish at
infinity, and $C_{A}(S)\subset C(S)$ the ideal of functions that vanish on
$A$. The map

$C_{A}(S)\ni f\mapsto f|(S\setminus A)\in C_{0}(S\setminus A)$

is an isometric isomorphism. Let furthermore $V^{*}\to S$ be the bundle dual
to $V$, endowed with the dual norm $|\,\,|^{*}$, and $\langle\,,\rangle$ the
duality pairing between $V^{*}$ and $V$. The tensor product

$C(V^{*})\otimes_{C(S)}M(S\setminus A)$

is the $C(S)$–module freely generated by $C(V^{*})\times M(S\setminus A)$,
modulo the the submodule generated by

(4.1)
$(fg_{1}+g_{2},m)-f(g_{1},m)-(g_{2},m),\quad(g,fm_{1}+m_{2})-f(g,m_{1})-(g,m_{2}),$

$f\in C(S)$, $g_{1},g_{2},g\in C(V^{*})$, $m_{1},m_{2},m\in M(S\setminus A)$.
Write $g\otimes m$ for the class of $(g,m)$.

######  Lemma 4.1.

Associating with $t=\sum_{1}^{k}g_{j}\otimes m_{j}\in
C(V^{*})\otimes_{C(S)}M(S\setminus A)$ the linear form

(4.2) $C_{A}(V)\ni\eta\mapsto\int_{S\setminus A}\sum_{j=1}^{k}\langle
g_{j},\eta\rangle\,dm_{j}\in\mathbb{C}$

defines an isomorphism

(4.3) $C(V^{*})\otimes_{C(S)}M(S\setminus
A)\overset{\approx}{\longrightarrow}C_{A}(V)^{\prime}$

of $C(S)$–modules.

###### Proof.

To see that different representations of $t$ produce the same linear form
(4.2) we have to check that (4.2) gives zero when the representations
correspond to the vectors in (4.1); and this is obvious. Thus (4.2) indeed
defines a homomorphism (4.3).

Next we verify that (4.3) is surjective. Let $l\in C_{A}(V)^{\prime}$. Suppose
$U\subset S$ is open and $V$ is trivial over a neighborhood of $\bar{U}$. Fix
$\xi_{1},\dots,\xi_{r}\in C(V)$ and $g_{1},\dots,g_{r}\in C(V^{*})$ that form
dual frames of $V,V^{*}$ over $U$. If $\chi\in C(S)$ is supported in $U$,
Riesz’s theorem, generalized [R87], implies that the linear forms

$C_{0}(S\setminus A)\approx C_{A}(S)\ni f\mapsto l(\chi
f\xi_{i}\big{)}\in\mathbb{C}$

can be represented as $l(\chi f\xi_{i})=\int_{S\setminus A}f\,dm_{i}$ with
$m_{i}\in M(S\setminus A)$. Hence

$l(\chi\eta)=\sum_{i=1}^{r}l\big{(}\chi\langle
g_{i},\eta\rangle\xi_{i}\big{)}=\sum_{i=1}^{r}\int_{S\setminus A}\langle
g_{i},\eta\rangle\,dm_{i},\qquad\eta\in C_{A}(V).$

If $\mathfrak{U}$ is a finite cover of $S$ consisting of $U$ as above,
$\chi=\chi_{U}$ form a subordinate partition of unity, and we choose
$r=r^{U}$, $\xi_{i}=\xi_{i}^{U}$, $g_{i}=g_{i}^{U}$, $m_{i}=m_{i}^{U}$
accordingly, then

$l(\eta)=\sum_{U\in\mathfrak{U}}l(\chi_{U}\eta)=\int_{S\setminus
A}\sum_{U\in\mathfrak{U}}\sum_{i=1}^{r^{U}}\langle
g_{i}^{U},\eta\rangle\,dm_{i}^{U},\qquad\eta\in C_{A}(V).$

Thus (4.3) sends $\sum_{U}\sum_{i}g_{i}^{U}\otimes m_{i}^{U}$ to $l$, and is
therefore surjective.

Finally, suppose that $t=\sum h_{j}\otimes m_{j}$ is in the kernel of (4.3).
Assume first that each $h_{j}$ is supported in an open $U$, and there are
$\xi_{1},\dots,\xi_{r}\in C(V)$, $g_{1},\dots,g_{r}\in C(V^{*})$ that form
dual frames over $U$. Then each $h_{j}$ is in the submodule generated by the
$g_{i}$, and $t$ can be written as $\sum_{i}g_{i}\otimes n_{i}$, with
$n_{i}\in M(S\setminus A)$ supported in $U$. Thus for any
$f_{1},\dots,f_{r}\in C_{A}(S)$ and $\eta=\sum_{i}f_{i}\xi_{i}\in C_{A}(V)$

$\sum_{i}\int_{S\setminus A}f_{i}\,dn_{i}=\int_{S\setminus A}\sum_{i}\langle
g_{i},\eta\rangle\,dn_{i}=0.$

Hence each $n_{i}=0$ and $t=0$. Now with a general $t=\sum_{j}h_{j}\otimes
m_{j}$ in the kernel of (4.3) and with a sufficiently fine partition of unity
$\chi_{U}$, $U\in\mathfrak{U}$, we have $t=\sum_{U}\chi_{U}t=0$ by what we
have already proved, since $\chi_{U}t$ is also in the kernel.

Next we define a bilinear pairing $P:B_{A}(V)\times
C_{A}(V)^{\prime}\to\mathbb{C}$. If $\eta\in B_{A}(V)$ and $l\in
C_{A}(V)^{\prime}$ corresponds, under (4.3), to $t=\sum_{j}g_{j}\otimes
m_{j}\in C(V^{*})\otimes_{C(S)}M(S\setminus A)$, set

(4.4) $P(\eta,l)=\int_{S\setminus A}\sum_{j}\langle
g_{j},\eta\rangle\,dm_{j}.$

Again, different representations of $t$ give the same integral in (4.4).

######  Lemma 4.2.

$B_{A}(V)\ni\eta\mapsto P(\eta,\cdot)\in C_{A}(V)^{\prime\prime}$ is an
isometric embedding, an extension of the canonical embedding $C_{A}(V)\to
C_{A}(V)^{\prime\prime}$.

###### Proof.

The statement about the extension is immediate from the definition of the
canonical embedding; which embedding is isometric. Next, fix $\eta\in
B_{A}(V)$ and $\sum_{j}g_{j}\otimes m_{j}\in
C(V^{*})\otimes_{C(S)}M(S\setminus A)$. Given $\varepsilon>0$, it follows from
Lusin’s theorem that there is a $\xi\in C_{A}(V)$ such that

$\sum_{j}\sup_{S}|g_{j}|^{*}\int_{S\setminus
A}|\xi-\eta|\,d|m_{j}|<\varepsilon\qquad\text{and}\qquad\sup_{S}|\xi|\leq\sup|\eta|.$

Hence $|P(\xi,l)-P(\eta,l)|=\big{|}\int_{S\setminus A}\sum_{j}\langle
g_{j},\xi-\eta\rangle\,dm_{j}\big{|}<\varepsilon$. Since
$||P(\xi,\cdot)||=\sup_{S}|\xi|$, we obtain
$||P(\eta,\cdot)||\leq\sup_{S}|\eta|$.

To prove the opposite inequality, let $s\in S\setminus A$ and $\delta_{s}$ the
Dirac measure at $s$. Choose a linear $\lambda:V_{s}\to\mathbb{C}$ such that

$|\lambda(v)|\leq|v|\quad\text{ for all }v\in V_{s},\quad\text{ with equality
if }v=\eta(s),$

and define $l(\xi)=\lambda\big{(}\xi(s)\big{)}$, $\xi\in C_{A}(V)$. Thus $l\in
C(V)^{\prime}$ corresponds under (4.3) to $g\otimes\delta_{s}$ with an
arbitrary $g\in C_{A}(V^{*})$ such that $g(s)=\lambda$. Hence
$|P(\eta,l)|=|\eta(s)|$. Since $||l||\leq 1$, we have
$||P(\eta,\cdot)||\geq|\eta(s)|$, and this being so for any $s\in S\setminus
A$, in fact $||P(\eta,\cdot)||\geq\sup_{S}|\eta|$. This completes the proof.

Identifying $B_{A}(V)$ with its image in $C_{A}(V)^{\prime\prime}$ allows us
to define a version of the Aron–Berner extension from $C_{A}(V)$ to
$B_{A}(V)$. If $\Gamma\subset B_{A}(V)$ is a ball with center in $C_{A}(V)$
and $f\in{\cal O}\big{(}\Gamma\cap C_{A}(V),\mathfrak{Y}\big{)}$ is bounded,
the extension $f^{\text{AB}}$ discussed in section 3 can be restricted to
$\Gamma$ to produce a holomorphic

(4.5) $f^{\text{ab}}:\Gamma\to\mathfrak{Y}^{\prime\prime},$

with the same sup norm. Similarly, if $T:C_{A}(V)^{\oplus k}\to\mathfrak{Y}$
is $k$–linear, the restriction to $B_{A}(V)^{\oplus k}$ of its Aron–Berner
extension $\bar{T}$ will be denoted $T^{\text{ab}}$.

######  Theorem 4.3.

In addition to $V\to S$, consider another topological complex vector bundle
$W\to S$, $\xi_{0}\in C_{A}(V)$, an open neighborhood $U\subset V$ of
${\xi_{0}(S)}$, and a ball $\Gamma\subset B_{A}(V)$ about $\xi_{0}$ such that
$\xi\in\Gamma$ implies $\overline{\xi(S)}\subset U$. Suppose a continuous
$H:U\to W$ maps fibers $U\cap V_{s}$ holomorphically into corresponding fibers
$W_{s}$. Then the map

$H^{\circ}:\Gamma\ni\xi\mapsto H\circ\xi\in B(W)$

is holomorphic.

If, furthermore, $H^{\circ}$ is bounded on $\Gamma\cap C_{A}(V)$, then

(4.6) $\big{(}H^{\circ}|\Gamma\cap C_{A}(V)\big{)}^{\text{ab}}=H^{\circ}.$

To prove Theorem 4.3 we need a lemma.

######  Lemma 4.4.

Suppose $J:V^{\oplus k}\to W$ is a continuous fiber map, fiberwise $k$–linear.
Define a $k$–linear operator $J^{\circ}:B_{A}(V)^{\oplus
k}\ni(\eta_{1},\dots,\eta_{k})\mapsto
J\circ(\eta_{1}\times\dots\times\eta_{k})\in B_{A}(W)$. Then

$(J^{\circ}|C_{A}(V)^{\oplus k})^{\text{ab}}=J^{\circ}.$

###### Proof.

In this proof we will use the topologies on $B_{A}(V),B_{A}(W)$ inherited from
the weak* topologies on $C_{A}(V)^{\prime\prime},C_{A}(W)^{\prime\prime}$. By
construction, these are induced by the pairings $P$ in (4.4), respectively the
corresponding $Q:B_{A}(W)\times C_{A}(W)^{\prime}\to\mathbb{C}$. First we show
that $J^{\circ}(\eta_{1},\dots,\eta_{k})\in B_{A}(W)$ depends continuously on
each $\eta_{j}\in B_{A}(V)$. Let’s prove for $j=1$; fix
$\eta_{2},\dots,\eta_{k}$. We need to show that
$Q\big{(}J^{\circ}(\eta_{1},\dots\,\eta_{k}),l\big{)}$, for any $l\in
C_{A}(W)^{\prime}$, depends continuously on $\eta_{1}$. The map

$K:V\ni v\mapsto J\big{(}v,\eta_{2}(s),\dots,\eta_{k}(s)\big{)}\in W,\qquad
v\in V_{s},\quad s\in S,$

is a bounded Borel homomorphism $V\to W$, with adjoint $K^{*}:W^{*}\to V^{*}$.
If $l$ corresponds to $\sum_{i}h_{i}\otimes m_{i}\in
C(W^{*})\otimes_{C(S)}M(S\setminus A)$, cf. (4.3), then

$\displaystyle
Q\big{(}J^{\circ}(\eta_{1},\dots,\eta_{k}),l\big{)}=\int_{S\setminus
A}\sum_{i}\langle h_{i},K\circ\eta_{1}\rangle\,dm_{i}=\int_{S\setminus
A}\sum_{i}\langle K^{*}\circ h_{i},\eta_{1}\rangle\,dm_{i}.$

The last expression is $l_{1}(\eta_{1})$ with a certain $l_{1}\in
C_{A}(V)^{\prime}$, and continuity follows.

In particular, $J^{\circ}$ is slightly continuous, as is
$(J^{\circ}|C_{A}(V)^{\oplus k})^{\text{ab}}$ by Lemma 3.1. Since the two
agree on $C_{A}(V)^{\oplus k}$, and $C_{A}(V)$ is dense in $B_{A}(V)$, they
agree everywhere.

###### Proof of Theorem 4.3.

By restricting to components of $S$, we can assume that $V$ and $W$ have
equidimensional fibers. Then Lemma 1.3 implies that $H^{\circ}$ even has a
holomorphic extension to a neighborhood of $\Gamma$ in $B(V)$.

Next we turn to (4.6). By translations we can arrange that $\xi_{0}=0$ and
$H^{\circ}(\xi_{0})=0$. In a neighborhood of the zero section of $V$ expand
$H$ in a uniformly convergent series $H=\sum_{k=0}^{\infty}H_{k}$, with
fiberwise $k$–homogeneous $H_{k}:V\to W$. Each $H_{k}$ is induced by a
symmetric $J:V^{\oplus k}\to W$ as in Lemma 4.4, $H_{k}(v)=J(v,\dots,v)$ for
$v\in V$. By the lemma and by (3.5), in a neighborhood of $0\in\Gamma$

$\big{(}H^{\circ}|\Gamma\cap
C_{A}(V)\big{)}^{\text{ab}}=\sum_{k}\big{(}H_{k}^{\circ}|\Gamma\cap
C_{A}(V)\big{)}^{\text{ab}}=\sum_{k}H_{k}^{\circ}=H^{\circ}.$

By analytic continuation therefore $\big{(}H^{\circ}|\Gamma\cap
C_{A}(V)\big{)}^{\text{ab}}=H^{\circ}$ on all of $\Gamma$.

We return to the notion of regularity, Definition 3.3.

######  Lemma 4.5.

The spaces $C_{A}(V)$ are regular.

###### Proof.

As a C* algebra, $C_{A}(S)$ is regular by [U87, Theorem 2.8]. (That theorem is
about regularity of an arbitrary bilinear operator $X\times X\to X$, when $X$
is a C* algebra. However, the justification given there applies to any
bilinear operator on $X$ with values in any Banach space.) It follows that
$C_{A}(S)^{\oplus q}$ is also regular for any $q\in\mathbb{N}$, because if
$S_{q}$ is the disjoint union of $q$ copies of $S$ and $A_{q}\subset S_{q}$ is
the disjoint union of $q$ copies of $A$, then $C_{A}(S)^{\oplus q}\approx
C_{A_{q}}(S_{q})$. Now $C_{A}(V)$ is a quotient of $C_{A}(S)^{\oplus q}$ for
some $q$. Indeed, if $\xi_{1},\dots,\xi_{q}\in C(V)$ span each fiber $V_{s}$,
then

$C_{A}(S)^{\oplus q}\ni(f_{1},\dots,f_{q})\mapsto\sum_{i=1}^{q}f_{i}\xi_{i}\in
C_{A}(V)$

is surjective. Since regularity passes down to quotients according to Ülger,
[U87, Corollary 2.4], $C_{A}(V)$ is indeed regular.

Putting this, Theorem 3.4, and Theorem 4.3 together we obtain:

######  Corollary 4.6.

Suppose that in Theorem 4.3 $H^{\circ}(\Gamma)$ is contained in a ball
$\Delta\subset B_{A}(W)$, centered at $H^{\circ}(\xi_{0})$. If $g\in{\cal
O}(\Delta\cap C_{A}(W),\mathfrak{Y})$ is bounded, then

(4.7) $(g\circ H^{\circ})^{\text{ab}}=g^{\text{ab}}\circ H^{\circ}.$

##  5 Extending germs in mapping spaces

The naturality property in Corollary 4.6 makes it possible to generalize the
Aron–Berner extension from Banach spaces to Banach manifolds that are mapping
spaces. Given $\mathfrak{x}_{0}\in C(S,X)$ and a closed $A\subset X$, as in
section 2, $\mathfrak{B}=\\{\mathfrak{x}\in
B(S,X):\mathfrak{x}|A=\mathfrak{x}_{0}|A\\},$ and $\mathfrak{C}$ is the
connected component of $\mathfrak{x}_{0}$ in$\\{\mathfrak{x}\in
C(S,X):\mathfrak{x}|A=\mathfrak{x}_{0}|A\\}$.

At this point we are extending germs of holomorphic functions only. Suppose
$\mathfrak{x}_{1}\in\mathfrak{C}$, $\mathfrak{Y}$ is a Banach space, and ${\bf
f}:(\mathfrak{C},\mathfrak{x}_{1})\to\mathfrak{Y}$ is a holomorphic germ. Let
$D\subset X\times X$ and $F:D\to TX$ be as in Lemma 2.1. Suppose
$\mathfrak{x}\in\mathfrak{C}$ is such that $\mathfrak{x}_{1}$ is contained in

(5.1)
$D_{A}(\mathfrak{x})=\\{\mathfrak{z}\in\mathfrak{B}:(\mathfrak{x}\times\mathfrak{z})(S)\text{
has compact closure in }D\\}\subset D(\mathfrak{x}),$

cf. (2.1). Thus $D_{A}(\mathfrak{x})$ is a coordinate neighborhood in
$\mathfrak{B}$, with local coordinate the restriction of
$\varphi_{\mathfrak{x}}$ of (2.5),

(5.2) $\psi_{\mathfrak{x}}:D_{A}(\mathfrak{x})\ni\mathfrak{z}\mapsto
F\circ(\mathfrak{x}\times\mathfrak{z})\subset B_{A}(\mathfrak{x}^{*}TX).$

Say, $\bf f$ is the germ of a bounded holomorphic function
$f:V\to\mathfrak{Y}$. The pullback $f\circ\psi_{\mathfrak{x}}^{-1}$ is
holomorphic in a neighborhood of $\psi_{\mathfrak{x}}(\mathfrak{x}_{1})\in
C_{A}(\mathfrak{x}^{*}TX)$; we can chose this neighborhood to be of form
$\Gamma\cap C_{A}(\mathfrak{x}^{*}TX)$ with $\Gamma\subset
B_{A}(\mathfrak{x}^{*}TX)$ a ball centered at
$\psi_{\mathfrak{x}}(\mathfrak{x}_{1})$. As we saw in section 4,
$f\circ\psi_{\mathfrak{x}}^{-1}$ then has an Aron–Berner type extension to a
holomorphic
$(f\circ\psi_{\mathfrak{x}}^{-1})^{\text{ab}}:\Gamma\to\mathfrak{Y}^{\prime\prime}$.
The germ of $(f\circ\psi_{\mathfrak{x}}^{-1})^{\text{ab}}$ at
$\psi_{\mathfrak{x}}(\mathfrak{x}_{1})$ is independent of which ball $\Gamma$
is chosen.

######  Lemma 5.1.

The germ of
$(f\circ\psi_{\mathfrak{x}}^{-1})^{\text{ab}}\circ\psi_{\mathfrak{x}}$ at
$\mathfrak{x}_{1}$ is independent of the choice of $\mathfrak{x}$.

###### Proof.

Suppose $\mathfrak{x}_{1}\in D_{A}(\mathfrak{y})$ with some
$\mathfrak{y}\in\mathfrak{C}$, and $g=f\circ\psi_{\mathfrak{y}}^{-1}$ is
bounded and holomorphic on $\Delta\cap C_{A}(\mathfrak{y}^{*}TX)$, where
$\Delta\subset B_{A}(\mathfrak{y}^{*}TX)$ is a ball about
$\psi_{\mathfrak{y}}(\mathfrak{x}_{1})$. The composition
$h=\psi_{\mathfrak{y}}\circ\psi_{\mathfrak{x}}^{-1}$ is holomorphic in a
neighborhood of $\psi_{\mathfrak{x}}(\mathfrak{x}_{1})\in
B_{A}(\mathfrak{x}^{*}TX)$; we can choose $\Gamma$ above contained in this
neighborhood, and so that $h(\Gamma)\subset\Delta$. (5.2) implies

$h(\xi)=F\circ\big{(}\mathfrak{y}\times(\exp\circ\xi)\big{)},\qquad\xi\in\Gamma,$

see (2.4). Denote the bundle projection $\mathfrak{x}^{*}TX\to S$ by $p$, and
for $v\in\mathfrak{x}^{*}TX$ in a neighborhood $U$ of
$\psi_{\mathfrak{x}}(\mathfrak{x}_{1})(S)$ set

$H(v)=F\big{(}\mathfrak{y}(pv),\exp(v)\big{)}\in\mathfrak{y}^{*}TX.$

This $H$ is continuous and fiberwise holomorphic, as in Theorem 4.3, and
$H^{\circ}$ of that theorem agrees with $h$ (on a sufficiently small
$\Gamma$). By Corollary 4.6, on $\Gamma$

$(f\circ\psi_{\mathfrak{y}}^{-1})^{\text{ab}}\circ\psi_{\mathfrak{y}}\circ\psi_{\mathfrak{x}}^{-1}=g^{\text{ab}}\circ
H^{\circ}=(g\circ
h)^{\text{ab}}=(f\circ\psi_{\mathfrak{x}}^{-1})^{\text{ab}},$

and indeed
$(f\circ\psi_{\mathfrak{y}}^{-1})^{\text{ab}}\circ\psi_{\mathfrak{y}}=(f\circ\psi_{\mathfrak{x}}^{-1})^{\text{ab}}\circ\psi_{\mathfrak{x}}$
in a neighborhood of $\mathfrak{x}_{1}$.

Thus we can make the following definition:

######  Definition 5.2.

If ${\bf f}:(\mathfrak{C},\mathfrak{x}_{1})\to\mathfrak{Y}$ is a holomophic
germ, then ${\bf
f}^{\text{ab}}:(\mathfrak{B},\mathfrak{x}_{1})\to\mathfrak{Y}^{\prime\prime}$
stands for its extension $({\bf
f}\circ\psi_{\mathfrak{x}}^{-1})^{\text{ab}}\circ\psi_{\mathfrak{x}}$. Here
$\mathfrak{x}\in\mathfrak{C}$ is arbitrary as long as $\mathfrak{x}_{1}\in
D_{A}(\mathfrak{x})$, and $\psi_{\mathfrak{x}}$ is the local coordinate (5.2).

##  6 Banach bundles and flat norms

In this section we introduce/review the notions in the title. The main result,
Theorem 6.3, is that over a simply connected Banach manifold a holomorphic
Banach bundle with a flat norm is trivial. Norms on vector bundles are also
called Finsler metrics; we have already discussed them for finite rank
bundles. We will not need the notion of general norms here, and will define
only flat norms.

######  Definition 6.1.

(a) A holomorphic Banach bundle is a holomorphic map $\pi:E\to M$ of complex
Banach manifolds, with each fiber $E_{x}=\pi^{-1}x$, $x\in M$, endowed with
the structure of a complex vector space. It is required that for each $x\in M$
there be a neighborhood $U\subset M$, a complex Banach space $(W,\|\,\,\|)$,
and a biholomorphism $g:\pi^{-1}U\to U\times W$ (local trivialization), that
for all $y\in U$ maps $E_{y}$ linearly to $\\{y\\}\times W$.

(b) A flat norm on $E\to M$ is a function $p:E\to[0,\infty)$ such that the
local trivializations $g$ above can be chosen to satisfy

$p(e)=||w||\qquad\text{if}\quad e\in\pi^{-1}U,\quad g(e)=(y,w).$

To describe the structure of flat normed holomorphic Banach bundles we need
the following result. If $W$ is a Banach space, $\text{End}\,W$ is the space
of continuous linear operators $W\to W$, endowed with the operator norm, and
$\text{GL}(W)\subset\text{End}\,W$ the set of invertibles.

######  Lemma 6.2.

Let $U$ be a connected open subset of a complex Banach space $Z$,
$(W,||\,\,||)$ another complex Banach space, and $F:U\to\text{GL}(W)$
holomorphic. If $||F(\zeta)w||=||w||$ for all $\zeta\in U$ and $w\in W$, then
$F$ is constant.

###### Proof.

It suffices to prove when $\dim Z=1$, the general case then follows by
restricting to one dimensional slices. So, let $Z=\mathbb{C}$. Denoting
derivative by a dot, we will show that $\dot{F}(\zeta)=0$ for arbitrary
$\zeta\in U$. To simplify notation, we assume $\zeta=0$, and denote $F(0)=P$,
$\dot{F}(0)=2Q$. Thus $F(\zeta)=P+2Q\zeta+\dots$ for small
$\zeta\in\mathbb{C}$. Let

$K(\tau)=1+\cos 2\pi\tau=1+(e^{2\pi i\tau}+e^{-2\pi i\tau})/2\geq
0,\qquad\tau\in\mathbb{R},$

be the first Fejér kernel. For $w\in W$ and small $\zeta$

$||(P+\zeta Q)w||=\big{\|}\int_{0}^{1}K(\tau)F(e^{2\pi
i\tau}\zeta)w\,d\tau\big{\|}\leq\int_{0}^{1}K(\tau)||F(e^{2\pi
i\tau}\zeta)w||\,d\tau=||w||=||Pw||.$

Since the left hand side is a subharmonic function of $\zeta$, the maximum
principle implies $||(P+\zeta Q)w||=||w||$, when $|\zeta|<r$ with some $r>0$.

Denote the norm on the dual space $W^{\prime}$ also by $||\,\,||$, and the
transpose of $P,Q$ by $P^{\prime},Q^{\prime}\in\text{End}\,W^{\prime}$; thus
$P^{\prime}$ is an isometry. Furthermore

(6.1) $||(P^{\prime}+\zeta Q^{\prime})u||=||u||,\qquad|\zeta|<r,\,u\in
W^{\prime},$

as well. Indeed, with $\langle\,,\rangle$ the pairing between $W^{\prime}$ and
$W$,

$||(P^{\prime}+\zeta
Q^{\prime})u||=\sup_{||w||=1}\big{|}\langle(P^{\prime}+\zeta
Q^{\prime})u,w\rangle\big{|}=\sup_{||w||=1}\big{|}\langle u,(P+\zeta
Q)w\rangle\big{|}\leq||u||=||P^{\prime}u||.$

Again, the left hand side is a subharmonic function of $\zeta$, and (6.1)
follows.

Suppose $u$ is an extreme point of the closed unit ball $\Delta\subset
W^{\prime}$. Then $P^{\prime}u\in\Delta$ is also an extreme point. Comparing
this with (6.1), we conclude $Q^{\prime}u=0$. Since in the weak* topology
$\Delta$ is compact, and by the Krein–Milman theorem it is the closed convex
hull of its extreme points, $Q^{\prime}v=0$ follows for all $v\in\Delta$.
Hence $\dot{F}(0)=2Q=0$.

######  Theorem 6.3.

Suppose $\pi:E\to M$ is a holomorphic Banach bundle over a connected, simply
connected base, and $p$ is a flat norm on $E$. Then $(E,p)$ is globally
trivial: there is a Banach space $(W,||\,\,||)$ and a biholomorphism $g:E\to
M\times W$ that maps each $E_{x}$ linearly to $\\{x\\}\times W$, and satisfies

$p(e)=||w||\qquad\text{if}\quad e\in E,\,g(e)=(x,w).$

This is a standard consequence of Lemma 6.2. The proof depends on the
following notion:

######  Definition 6.4.

Suppose $U\subset M$ is connected open and $E|U$ admits a trivialization
$g:E|U\to U\times W$ as in Definition 6.1b. A section $h$ of $E|U$ is called
horizontal if $g\circ h:U\to U\times W$ is a constant section. (Lemma 6.2
implies that horizontality of $h$ is independent of the choice of the local
trivialization $g$.) A section $h$ over a general open subset of $M$ is called
horizontal if it is locally such.

###### Proof of Theorem 6.3.

Define a Hausdorff topology on $E$ by declaring that a neighborhood basis of
$e\in E$ consists of sets of form $h(U)$, where $U\subset M$ is a neighborhood
of $\pi(e)$ and $h$ is a horizontal section of $E|U$. In this topology
$\pi:E\to M$ is a covering, which must be trivial since $M$ is connected and
simply connected. This means that through every $e\in E$ there passes a unique
global horizontal section $h_{e}$. Fix an $a\in M$ and let
$(W,||\,||)=(E_{a},p|E_{a})$. This is a Banach space, and

$g:E\ni e\mapsto\big{(}\pi(e),h_{e}(a)\big{)}\in M\times W$

is the global trivialization sought.

If $E\to M$ is a holomorphic Banach bundle with a flat norm $p$ over a
connected base, the biduals of the fibers $E_{x}$ constitute a holomorphic
Banach bundle $E^{\prime\prime}\to M$, on which the biduals of the norms
$p|E_{x}$ define a flat norm $p^{\prime\prime}$. Local trivializations $g$ of
$(E,p)$ as in Definition 6.1b give rise to the corresponding local
trivializations $g^{\prime\prime}$ of $(E^{\prime\prime},p^{\prime\prime})$:
$g^{\prime\prime}(y)$ is just the second transpose of $g(y)$, $y\in U$.

These considerations lead to the Aron–Berner extension of germs of holomorphic
sections of holomorphic Banach bundles. Suppose that $\mathfrak{X}$ is a
Banach space, $B\subset\mathfrak{X}$ is a ball centered at $x\in\mathfrak{X}$,
$B^{\prime\prime}\subset\mathfrak{X}^{\prime\prime}$ is the bidual ball, and
$(E,p)\to B^{\prime\prime}$ is a holomorphic Banach bundle with a flat norm.
Under the trivialization $g$ of Theorem 6.3, a bounded holomorphic section $f$
of $E|B$ induces a bounded holomorphic section $g\circ f$ of $B\times W\to B$,
i.e., a bounded $\phi\in{\cal O}(B,W)$. The corresponding trivialization
$g^{\prime\prime}$ of $(E^{\prime\prime},p^{\prime\prime})$ and the
Aron–Berner extension $\phi^{\text{AB}}\in{\cal
O}(B^{\prime\prime},W^{\prime\prime})$, in turn, induce a bounded holomorphic
section $f^{\text{AB}}$ of $(E^{\prime\prime},p^{\prime\prime})$. Lemma 6.2
implies that any another norm respecting trivialization $E\to
B^{\prime\prime}\times W$ is of form
$g_{1}=(\text{id}_{B^{\prime\prime}}\times T)\circ g$, with
$T\in\text{End}\,W$. This means that the corresponding trivialization of
$E^{\prime\prime}$ is
$g^{\prime\prime}_{1}=(\text{id}_{B^{\prime\prime}}\times
T^{\prime\prime})\circ g^{\prime\prime}$. Since the corresponding
$\phi_{1}=T\circ\phi$, by Theorem 3.4
$\phi_{1}^{\text{AB}}=T^{\prime\prime}\circ\phi^{\text{AB}}$; hence
$f^{\text{AB}}$ is independent of the choice of $g$.

More generally, given a flat holomorphic Banach bundle $(E,p)$ over some open
set in $\mathfrak{X}^{\prime\prime}$, and a germ $\bf f$ of a local
holomorphic section of $E|\mathfrak{X}$ at some $x\in\mathfrak{X}$, the same
prescription defines the extension ${\bf f}^{\text{AB}}$, a germ of a section
of $E^{\prime\prime}$. One can even consider bundles over more general
manifolds. Combining the above construction with the discussion of mapping
spaces $\mathfrak{C}\subset\mathfrak{B}$ in section 5, given a flat
holomorphic Banach bundle $(E,p)\to\mathfrak{B}$, with any $E|\mathfrak{C}$
valued holomorphic germ $\bf f$ at some $\mathfrak{x}\in\mathfrak{C}$ we can
associate its Aron–Berner–type extension ${\bf f}^{\text{ab}}$, a holomorphic
germ at $\mathfrak{x}$, valued in $E^{\prime\prime}$.

##  7 Analytic continuation and extension

Consider a holomorphic Banach bundle $E\to M$ over a connected base, and let
${\cal E}\to M$ denote the sheaf of $E$ valued holomorphic germs. Suppose
$a<b$ are real numbers and $r:[a,b]\to M$ a path (that is, a continuous map).

######  Definition 7.1.

Given a germ ${\bf f}\in{\cal E}$ at $r(a)$, its analytic continuation along
the path $r$ is a continuous lift $[a,b]\ni t\mapsto{\bf f}_{t}\in{\cal
E}_{r(t)}$ of $r$ such that ${\bf f}_{a}=\bf f$.

Recall that continuity at $t\in[a,b]$ means that for all $\tau\in[a,b]$
sufficiently close to $t$ the ${\bf f}_{\tau}$ are germs of the same local
section of $E$ about $r(t)$. For this reason analytic continuation can be
defined by a partition $a=t_{0}<t_{1}<\dots<t_{k}=b$, open neighborhoods
$U_{j}\subset M$ of $r[t_{j-1},t_{j}]$, $j=1,\dots,k$, and holomorphic
sections $g_{j}$ of $E|U_{j}$ such that $g_{j}=g_{j+1}$ on a neighborhood of
$r(t_{j})$, and $\bf f$ is the germ of $g_{1}$ at $r(a)$.

This shows that analytic continuation along $r$ is unique; and that if the
path is slightly perturbed, keeping its endpoints fixed, then the continuation
of $\bf f$ at $r(b)$ is not going to change. It follows that if a germ ${\bf
f}\in{\cal E}_{x}$ admits analytic continuation along every path in $M$
starting at $x$, then the analytic continuation at $y\in M$ depends only on
the homotopy class of the path connecting $x$ and $y$ along which we continue.
In particular, if $M$ is also simply connected, then the analytic
continuations define a holomorphic section $f$ of $E$: $f(y)\in E_{y}$ will be
the value at $y$ of the analytic continuation from $x$ to $y$.

Later on we will need the following.

######  Lemma 7.2.

Suppose $P:M\to N$ is a holomorphic and open map of complex Banach manifolds,
$F\to N$ is a holomorphic Banach bundle, $[a,b]\ni t\mapsto x_{t}\in M$ is a
path, and ${\bf g}_{t}$ are $F$ valued holomorphic germs at $P(x_{t})\in N$.
Let ${\bf k}_{t}$ be the pullback of ${\bf g}_{t}$ along $P$ to $x_{t}$,
holomorphic $P^{*}F$ valued germs. If ${\bf k}_{t}$ depend continuously on
$t$, then so do ${\bf g}_{t}$.

###### Proof.

To prove continuity at some $t\in[a,b]$, let $V\subset N$ be a neighborhood of
$P(x_{t})$ and $g$ a holomorphic section of $F|V$ whose germ at $P(x_{t})$ is
${\bf g}_{t}$. Thus $g\circ P$ is a holomorphic section of $P^{*}F|P^{-1}V$
whose germ at $x_{t}$ is ${\bf k}_{t}$. For $\tau\in[a,b]$ close to $t$, ${\bf
k}_{\tau}$ is the germ of $g\circ P$ at $x_{\tau}$. As $P$ is open, this
implies that the germ of $g$ at $P(x_{\tau})$ is ${\bf g}_{\tau}$; which shows
that $t\mapsto g_{t}$ is indeed continuous.

In what follows, when we speak of analytic continuation of a germ ${\bf
f}\in{\cal E}_{x}$ along an arbitrary path, we of course mean paths that start
at $x$. Let $p$ be a flat norm on $E$.

######  Definition 7.3.

We say that a germ ${\bf f}\in{\cal E}_{x}$ admits bounded analytic
continuation along every path if first, it admits analytic continuations
$t\mapsto{\bf f}_{t}\in{\cal E}_{r(t)}$ along every path $r$ and second, there
is a bound $C$ such that for all such continuations $p\big{(}{\bf
f}_{t}(r(t))\big{)}\leq C$.

If $M$ is simply connected, bounded analytic continuation along every path
simply means that $\bf f$ extends to a holomorphic section $f$ of $E$ and
$p(f)$ is bounded.

Let us return to mapping spaces of finite dimensional connected complex
manifolds $X$. As earlier, given a compact Hausdorff space $S$, a closed
$A\subset S$, and a continuous $\mathfrak{x}_{0}:S\to X$, we consider the
spaces

(7.1)
$\mathfrak{B}=B(S,X,A,\mathfrak{x}_{0})\supset\mathfrak{C}=C(S,X,A,\mathfrak{x}_{0});$

$\mathfrak{B}=\\{\mathfrak{x}\in B(S,X):\mathfrak{x}|A=\mathfrak{x}_{0}|A\\}$
and $\mathfrak{C}$ is the connected component of $\mathfrak{x}_{0}$ in
$\mathfrak{B}\cap C(S,X)$. Let furthermore $E\to\mathfrak{B}$ be a holomorphic
Banach bundle. In section 6 we associated with an $E|\mathfrak{C}$ valued
holomorphic germ $\bf f$ at some $\mathfrak{y}\in\mathfrak{C}$ its Aron–Berner
type extension ${\bf f}^{\text{ab}}$, an $E^{\prime\prime}$ valued germ at
$\mathfrak{y}$.

######  Lemma 7.4.

If $t\mapsto{\bf f}_{t}$ is the analytic continuation of $\bf f$ along a path
in $\mathfrak{C}$, then $t\mapsto{\bf f}_{t}^{\text{ab}}$ is the continuation
of ${\bf f}^{\text{ab}}$ along the same path.

###### Proof.

It suffices to prove when the path is contained in a coordinate neighborhood.
Then we might as well assume that $\mathfrak{C}$ is a ball in a Banach space
$\mathfrak{X}$, $\mathfrak{B}\subset\mathfrak{X}^{\prime\prime}$ is a ball in
a subspace, $E\to\mathfrak{B}$ is trivial, and so $E$ and $E^{\prime\prime}$
valued germs $\bf f$ correspond to germs $\bf g$ valued in fixed Banach spaces
$\mathfrak{Y},\mathfrak{Y}^{\prime\prime}$. Now a special case of a result of
Zalduendo, that concerns the Aron–Berner extension from a Banach space to its
bidual, [Z90, Section 2, Definition 3, Theorem 1], implies that for
$\mathfrak{Y}$ valued germs $\bf g$ the analytic continuation of ${\bf
g}^{\text{ab}}$ is given by ${\bf g}_{t}^{\text{ab}}$, and this proves the
lemma.

The following theorem generalizes Theorem 0.1:

######  Theorem 7.5.

Suppose that a holomorphic Banach bundle $E\to\mathfrak{B}$ is endowed with a
flat norm $p$ (Definition 6.1). Assume that an $E|\mathfrak{C}$ valued
holomorphic germ $\bf f$ at some $\mathfrak{x}\in\mathfrak{C}$ admits bounded
analytic continuation along any path in $\mathfrak{C}$. Then the
$E^{\prime\prime}$ valued germ ${\bf f}^{\text{ab}}$ admits analytic
continuation along any path in $\mathfrak{B}$, with the same bound as the
continuations within $\mathfrak{C}$.

The proof will be given in section 9, after preliminary work in section 8.

##  8 The key construction

The key to the proof of Theorem 7.5 is a construction of a family of
holomorphic maps of balls into $X$, with the help of which we can pull back
analytic continuation along certain paths in $\mathfrak{B}$ to continuation in
a Banach space. Endow $X$ with a metric $d$, then, as in section 2, metrize
the space $B(S,X)$ by the metric

$d_{S}(\mathfrak{x},\mathfrak{y})=\sup\\{d\big{(}\mathfrak{x}(s),\mathfrak{y}(s)\big{)}:\,s\in
S\\}.$

######  Lemma 8.1.

Given $\mathfrak{x}\in C(S,X)$ and $\varepsilon>0$, there is a $\delta>0$ with
the following property. Suppose $S=\coprod_{0}^{J}S_{j}$ is a Borel partition,
$S_{0}=A$ is closed, $a<b$ are real numbers, and $[a,b]\ni
t\mapsto\mathfrak{y}_{t}\in B(S,X)$ is a continuous path. Suppose furthermore
that $\mathfrak{y}_{t}=\mathfrak{x}$ on $S_{0}$ and $\mathfrak{y}_{t}$ is
constant on $S_{j}$, $j=1,\dots,J$, $t\in T$. If
$d_{S}(\mathfrak{x},\mathfrak{y}_{a})<\delta$, then there are a (finite rank)
normed topological complex vector bundle $\pi:(V,|\,\,|)\to S$ (cf. section 4)
with unit ball bundle $U=\\{v\in V:|v|<1\\}\to S$, a $\mathfrak{v}\in
B_{A}(V)$, $\sup_{S}|\mathfrak{v}|<1$, and a continuous $\Psi:[a,b]\times U\to
X\times\mathbb{C}$ such that for every $t\in[a,b]$ and $s\in S$
(a) $\Psi$ maps the fibers $\\{t\\}\times U_{s}$ biholomorphically on open
subsets of $X\times\mathbb{C}$;
(b) the zero vector $0_{s}\in V_{s}$ satisfies
$\Psi(t,0_{s})=\big{(}\mathfrak{x}(s),0\big{)}$;
(c) with $\text{pr}_{X}:X\times\mathbb{C}\to X$ the projection,
$\text{pr}_{X}\circ\Psi(t,\mathfrak{v})=\mathfrak{y}_{t}$;
(d) $d\big{(}\mathfrak{x}(s),\text{pr}_{X}(\Psi(a,v))\big{)}<\varepsilon$ for
$v\in U_{s}$.

The proof takes some preparation. Most of the construction will revolve around
real analyticity. Let $Y$ be a connected real analytic manifold. In what
follows, we will drop ‘real’ from ‘real analytic’. By a theorem of Grauert,
$Y$ can be properly and analytically embedded in some Euclidean space [G58];
accordingly, we will assume $Y\subset\mathbb{R}^{N}$ is a closed analytic
submanifold. For any $x\in\mathbb{R}^{N}$ in some neighborhood $R$ of $Y$
there is a unique nearest point $r(x)$ in $Y$; the map $r:R\to Y$ is an
analytic retraction.

Fix a topological space $T$ and a compact interval $I\subset\mathbb{R}$. We
will say that a map $g:T\times I\to Y$ is partly analytic, $g\in
C^{0\omega}(T\times I,Y)$, if $T\times I\subset T\times\mathbb{C}$ has a
neighborhood $O$ to which $g$ extends as a continuous function
$\tilde{g}:O\to\mathbb{C}^{N}$ whose restrictions $\tilde{g}(t,\cdot)$ are
holomorphic, $t\in T$. One can check that partial analyticity is independent
of which embedding $Y\subset\mathbb{R}^{N}$ is chosen, and it does not even
matter if $Y$ is closed in $\mathbb{R}^{N}$, or only in some open subset of
$\mathbb{R}^{N}$. It is clear that the composition of the partly analytic $g$
with an analytic map $Y\to Y_{1}$ produces a partly analytic map.

######  Lemma 8.2.

Suppose $M,P$ are finite dimensional complex manifolds, and $g\in
C^{0\omega}\big{(}(T\times M)\times I,P\big{)}$ is such that
$g(t,\cdot,q):M\to P$ is holomorphic for each $t\in T$, $q\in I$. Then
$T\times M\times I\subset T\times M\times\mathbb{C}$ has a neighborhood $O$ to
which $g$ extends as a continuous $f:O\to P$, whose restrictions
$f(t,\cdot,\cdot)$ are holomorphic for each $t\in T$. If
$O_{t,z}=\\{q\in\mathbb{C}:(t,z,q)\in O\\}$ are convex (that can always be
achieved at the price of shrinking $O$), such extension $f$ is unique.

###### Proof.

Assume first that $M$, $P$ are open subsets of some $\mathbb{C}^{m}$,
$\mathbb{C}^{p}$. With $w_{j}$, $\zeta_{h}$ the complex coordinates on
$\mathbb{C}^{p}$, respectively $\mathbb{C}^{2p}$, consider the embedding
$\alpha$ of $\mathbb{C}^{p}$ (and so of $P$) into
$\mathbb{R}^{2p}\subset\mathbb{C}^{2p}$, and its left inverse $\beta$,

$\alpha(w)=(\text{Re}\,w_{1},\text{Im}\,w_{1},\dots,\text{Re}\,w_{p},\text{Im}\,w_{p}),\qquad\beta(\zeta)=(\zeta_{1}+i\zeta_{2},\dots,\zeta_{2p-1}+i\zeta_{2p}).$

By definition, $\alpha\circ g$ extends to a continuous
$\tilde{g}:O\to\mathbb{C}^{N}$ whose restrictions $\tilde{g}(t,z,\cdot)$ are
holomorphic. It follows that, possibly after shrinking $O$,
$f=\beta\circ\tilde{g}:O\to P$ extends $g$ and has holomorphic restrictions
$f(t,z,\cdot)$; we can also arrange that the $O_{t,z}$ are convex. That even
the restrictions $f(t,\cdot,\cdot)$ are holomorphic will follow by Morera’s
and Osgood’s theorems once we show, denoting the coordinates on
$\mathbb{C}^{m}$ by $z_{1},\dots,z_{m}$, that each $(t,z,q)\in O$ has a
neighborhood $G\subset U$ such that $\int_{\Gamma}f(t,z,q)\,dz_{k}=0$ whenever
$\Gamma\subset M$ is a smooth loop contained in a complex line,
$\\{t\\}\times\Gamma\times\\{q\\}\subset G$, and $k=1,\dots,m$. But this is
clear, as the integral is a holomorphic function of $q$, vanishing when $q\in
I$.

The extension $f$ is unique by the uniqueness theorem for holomorphic
functions.

For a general $P$, cover $T\times M\times I$ by subsets $T_{i}\times
M_{i}\times I_{i}$ that $g$ maps into coordinate neighborhoods on $P$, $T_{i}$
open, $M_{i}\subset M$ coordinate neighborhoods, $I_{i}\subset I$ compact
intervals. We have just constructed extensions $f_{i}:O_{i}\to P$ to
neighborhoods $O_{i}$ of $T_{i}\times M_{i}\times I_{i}$. By uniqueness the
$f_{i}$ are compatible and produce the extension $f$ claimed.

######  Lemma 8.3.

If $T$ is compact and $L\subset I$ is finite, any $f\in C(T\times I,Y)$ can be
approximated arbitrarily closely (in the uniform topology) by $g\in
C^{0\omega}(T\times I,Y)$ such that $f=g$ on $T\times L$.

###### Proof.

We will avail ourselves of the following tools. There is a linear operator of
interpolation, $\Lambda$, from the space $C(L,\mathbb{R}^{N})$ of all
functions $L\to\mathbb{R}^{N}$ to the space of polynomials
$\mathbb{R}\to\mathbb{R}^{N}$ such that $(\Lambda\phi)|L=\phi$ for $\phi\in
C(L,\mathbb{R}^{N})$. Out of $\Lambda$ we construct

$\bar{\Lambda}:C(T\times L,\mathbb{R}^{N})\to C^{0\omega}(T\times
I,\mathbb{R}^{N}),\qquad(\bar{\Lambda}\varphi)(t,\cdot)=\Lambda\big{(}\varphi(t,\cdot)\big{)},$

continuous between the uniform topologies on $C(T\times L,\mathbb{R}^{N})$ and
$C^{0\omega}(T\times I,\mathbb{R}^{N})$.

Furthermore, any $\varphi\in C(T\times I,\mathbb{R}^{N})$ can be approximated
uniformly by functions in $C^{0\omega}(T\times I,\mathbb{R}^{N})$. (For this
latter, extend $\varphi$ to a compactly supported continuous
$\tilde{\varphi}:T\times\mathbb{R}\to\mathbb{R}^{N}$. The approximating
functions to $\varphi$ can be obtained by convolutions, for example, with the
Poisson kernel:

$\varphi_{\varepsilon}(t,x)=\frac{\varepsilon}{\pi}\int_{\mathbb{R}}\dfrac{\tilde{\varphi}(t,y)\,dy}{|x-y|^{2}+\varepsilon^{2}},\qquad(t,x)\in
T\times I,\quad\varepsilon>0.$

Then $\varphi_{\varepsilon}\in C^{0\omega}(T\times I,\mathbb{R}^{N})$ tend
uniformly to $\varphi$ as $\varepsilon\to 0$.)

To construct $g$ of the lemma, first uniformly approximate $f$ by $\phi\in
C^{0\omega}(T\times I,\mathbb{R}^{N})$. Next let
$\psi=\phi+\bar{\Lambda}\big{(}(f-\phi)|T\times L\big{)}\in
C^{0\omega}(T\times I,\mathbb{R}^{N})$, so that $\psi|T\times L=f|T\times L$.
If the approximation by $\phi$ was good enough, then $\phi$ and $\psi$ map
$T\times I$ into the retract neighborhood $R$, and $g=r\circ\psi$ is the
approximation of $f=r\circ f$ sought.

We return to the set up of Lemma 8.1. Let $I\subset\mathbb{R}$ be a compact
interval and $q_{0},\dots,q_{J}\in I$ distinct points.

######  Lemma 8.4.

Given $\mathfrak{x}\in C(S,X)$ and $\varepsilon>0$, there is a $\delta>0$ with
the following property. Suppose $S=\coprod_{0}^{J}S_{j}$ is a Borel partition,
$S_{0}$ is closed, $a<b$ are real numbers, and $[a,b]\ni
t\mapsto\mathfrak{y}_{t}\in B(S,X)$ is a continuous path. Suppose furthermore
that $\mathfrak{y}_{t}=\mathfrak{x}$ on $S_{0}$, and each $\mathfrak{y}_{t}$
is constant on $S_{j}$, $j=1,\dots,J$. If
$d_{S}(\mathfrak{x},\mathfrak{y}_{a})<\delta$, then there is a $\psi\in
C^{0\omega}\big{(}([a,b]\times S)\times I,X\big{)}$ such that
(a) $\psi(t,s,q_{0})=\mathfrak{x}(s)$ if $t\in[a,b]$, $s\in S$;
(b) $\psi(t,s,q_{j})=\mathfrak{y}_{t}(s)$ if $t\in[a,b]$, $s\in S_{j}$,
$j=1,\dots,J$;
(c) $d\big{(}\mathfrak{x}(s),\psi(a,s,q)\big{)}<\varepsilon$ if $s\in S$,
$q\in I$.

###### Proof.

Embed $X$ as a closed real analytic submanifold of some $\mathbb{R}^{N}$, with
$r:R\to X$ an analytic retraction of a neighborhood $R\subset\mathbb{R}^{N}$.
We can assume that the metric $d$ of $X$ is the restriction of the Euclidean
metric; and also that $a=0$ and $\varepsilon<1$. Choose $\delta<\varepsilon/2$
so that the $\delta$–neighborhood of

$\\{x\in X:|x|\leq 1+\sup_{S}|\mathfrak{x}|\\}\subset\mathbb{R}^{N}$

is contained in the retract neighborhood $R$; and over this
$\delta$–neighborhood the retraction satisfies $|r-\text{id}|<\varepsilon/2$.
Given $\mathfrak{y}_{t}$, we will construct $\psi$ in several steps. Consider

$C=\big{(}S\times\\{q_{0}\\}\big{)}\cup\bigcup_{j=1}^{J}\big{(}\bar{S}_{j}\times\\{q_{j}\\}\big{)},\qquad
C_{1}=[0,b]\times C,\qquad C_{2}=\\{0\\}\times S\times I;$

$C_{1},C_{2}$ are closed subsets of $[0,b]\times S\times I$. Define
$\psi_{1}:C_{1}\to X$ by

$\psi_{1}(t,s,q_{j})=\begin{cases}\mathfrak{x}(s)&\text{if }j=0,\quad s\in
S\\\ \mathfrak{y}_{t}(S_{j})&\text{if }j\geq 1,\quad
s\in\bar{S}_{j}.\end{cases}$

In particular

(8.1) $d\big{(}\mathfrak{x}(s),\psi_{1}(0,s,q)\big{)}\leq
d_{S}(\mathfrak{x},\mathfrak{y}_{0})<\delta\qquad\text{if}\quad(0,s,q)\in
C_{1}\cap C_{2}.$

Let $B_{s}\subset R$ denote the closed ball of radius $\delta$ about
$\mathfrak{x}(s)$. Extend $\psi_{1}|C_{1}\cap C_{2}$ to a continuous function
$\psi_{2}:C_{2}\to\mathbb{R}^{N}$, then for $(s,q)\in S\times I$ define
$\psi_{3}(0,s,q)\in B_{s}$ as the closest point to $\psi_{2}(0,s,q)$. Thus
$\psi_{3}:C_{2}\to R$ is continuous and by (8.1)
$\psi_{1}=\psi_{2}=\psi_{3}=r\circ\psi_{3}$ on $C_{1}\cap C_{2}$. Hence

$\psi_{4}=\begin{cases}\psi_{1}&\text{on }C_{1}\\\ r\circ\psi_{3}&\text{on
}C_{2}\end{cases}$

defines a continuous function $C_{1}\cup C_{2}\to X$, which extends to a
continuous $\psi_{5}:G\to X$, with some neighborhood $G$ of $C_{1}\cup C_{2}$.
Choose a neighborhood $G_{1}\subset S\times I$ of $C$ such that $[0,b]\times
G_{1}\subset G$, and let a continuous $\chi:S\times I\to[0,1]$ be supported in
$G_{1}$, equal to $1$ on $C_{1}\cup C_{2}$. Then
$\psi_{6}(t,s,q)=\psi_{5}\big{(}t\chi(s,q),s,q\big{)}$ defines a continuous
function $[0,b]\times S\times I\to X$, that agrees with $\psi_{1}$ on $C_{1}$
and with $r\circ\psi_{3}$ on $C_{2}$. Requirements (a,b) of the lemma hold for
$\psi_{6}$ by the definition of $\psi_{1}$, and (c) follows because
$(0,s,q)\in C_{2}$ when $s\in S$, $q\in I$, and so

$d\big{(}\mathfrak{x}(s),\psi_{6}(0,s,q)\big{)}=d\big{(}\mathfrak{x}(s),r(\psi_{3}(0,s,q))\big{)}\\\
\leq
d\big{(}\mathfrak{x}(s),\psi_{3}(0,s,q)\big{)}+d\big{(}\psi_{3}(0,s,q),r(\psi_{3}(0,s,q))\big{)}<\delta+\varepsilon/2<\varepsilon.$

A sufficiently good uniform approximation $\psi\in
C^{0\omega}\big{(}([a,b]\times S)\times I,X\big{)}$ of $\psi_{6}$ such that
$\psi(t,s,q_{j})=\psi_{6}(t,s,q_{j})$ for all $t,s,j$, as guaranteed by Lemma
8.3, will then do.

###### Proof of Lemma 8.1.

To start, let $D\subset X\times X$ be a neighborhood of the diagonal and
$F:D\to TX$ as in Lemma 2.1, and $\exp$ as in (2.4). Possibly after shrinking
$X$ we can arrange that $F$ and therefore $\exp$ are analytic.

Consider the holomorphic vector bundle $\text{Hom}\to X\times X$ whose fibers
are $\text{Hom}_{x,y}=\text{Hom}\,(T_{x}X,T_{y}X)$, and the open subset
$\text{Iso}\subset\text{Hom}$ consisting of isomorphisms. The fiber bundle
$p:\text{Iso}\to X\times X$ has a canonical analytic section, $A$, over $D$:
this is so, because for fixed $x$ the tangent spaces to $T_{x}X$ are
canonically isomorphic, and the differential of $F(x,\cdot)$ sets up
isomorphisms between $T_{x}X$, resp. $T_{y}X$ on the one hand, and the tangent
spaces to $T_{x}X$ at $F(x,x),F(x,y)$ on the other.
$A(x,y)\in\text{Hom}\,_{x,y}$ is then the composition of these isomorphisms or
their inverses.

Let $I=[-1,1]$ and $q_{0}=0,q_{1},\dots,q_{J}\in I$ distinct points. We will
show that the $\delta$ that Lemma 8.4. provides works for Lemma 8.1 as well.
Given $a,b,\mathfrak{y}_{t}$ as in Lemma 8.1, let $\psi\in
C^{0\omega}\big{(}([a,b]\times S)\times I,X\big{)}$ be the map that Lemma 8.4
constructs. With $\psi_{0}=\psi(a,\cdot,0):S\to X$ consider the sum of
$\psi_{0}^{*}TX\to S$ and the trivial line bundle,

$\pi:V=(\psi_{0}^{*}TX)\times\mathbb{C}\to S,$

into which $(\psi_{0}^{*}TX)\times I$ is embedded.

Let furthermore $\tilde{\psi}(t,s,q)=\psi_{0}(s)=\psi(a,s,0)$. The vector
bundle $\tilde{\psi}^{*}TX$ is isomorphic to $\psi^{*}TX$, because
$\psi,\tilde{\psi}:[a,b]\times S\times I\to X$ are homotopic. We use this
isomorphism to construct a $C^{0\omega}$ map

(8.2) $\theta:([a,b]\times S)\times I\to\text{Iso},\qquad
p\big{(}\theta(t,s,q)\big{)}=\big{(}\psi_{0}(s),\psi(t,s,q)\big{)},$

as follows. An isomorphism of the bundles translates to a continuous map
$\theta_{0}$ as in (8.2), that, by Lemma 8.3, can be approximated by
$\theta_{1}\in C^{0\omega}\big{(}([a,b]\times S)\times I,\text{Iso}\big{)}$.
Let $p\circ\theta_{1}=\alpha\times\beta$, with $\alpha,\beta:[a,b]\times
S\times I\to X$ close to $\tilde{\psi}$, resp. $\psi$. Then

$\theta=\big{(}A\circ(\beta\times\psi)\big{)}\,\theta_{1}\,\big{(}A\circ(\tilde{\psi}\times\alpha)\big{)}\in
C^{0\omega}\big{(}([a,b]\times S)\times I,\text{Iso}\big{)}$

satisfies (8.2). (Above the values of $\theta$ are products of three linear
transformations.)

With $\varpi:\psi_{0}^{*}TX\to S$ the bundle projection and
$\Xi\subset\psi_{0}^{*}TX$ a suitable neighborhood of the zero section,

(8.3) $\phi:([a,b]\times\Xi)\times
I\ni(t,\xi,q)\mapsto\big{(}\exp(\theta(t,\varpi\xi,q)\xi),q\big{)}\in
X\times\mathbb{C}$

defines a $C^{0\omega}$ map that is holomorphic along the fibers of $\varpi$.
We have

(8.4) $\phi(t,\xi_{0},q)=\big{(}\psi(t,s,q),q\big{)}\qquad\text{when
$\xi_{0}\in(\psi_{0}^{*}TX)_{s}$ is the zero vector,}$

since then $\theta(t,\varpi\xi_{0},q)\xi_{0}\in T_{\psi(t,s,q)}X$ is the zero
vector.

Locally in the base, the bundle $\psi_{0}^{*}TX$ is a direct product, and
$\Xi$ contains a neighborhood of the zero section that is a product of a piece
of $S$ with a complex manifold. Lemma 8.2 therefore gives, possibly after
shrinking $\Xi$, an extension of $\phi$ to a neighborhood
$O\subset[a,b]\times\Xi\times\mathbb{C}\subset[a,b]\times V$ of
$[a,b]\times\Xi\times I$. The extension

$\Phi=\Phi_{1}\times\Phi_{2}:O\to X\times\mathbb{C}$

is continuous, and holomorphic along the fibers $O_{t,s}=O\cap(\\{t\\}\times
V_{s})$. By (8.3) $\Phi_{2}(t,\xi,q)=q$. We embed $S$ into $\psi_{0}^{*}TX$ as
the zero section, and claim that—possibly after shrinking $O$ to a smaller
neighborhood of $[a,b]\times S\times I\subset[a,b]\times V$—the restrictions
$\Phi|O_{t,s}$ map biholomorphically on open subsets of $X\times\mathbb{C}$.

Indeed, fix a zero vector $\xi_{0}\in T_{\psi_{0}(s)}X$ and $q\in I$.
According to the splitting $V=(\psi_{0}^{*}TX)\times\mathbb{C}$, the tangent
space $T_{(t,\xi_{0},q)}O_{t,s}$ also splits

(8.5) $T_{(t,\xi_{0},q)}O_{t,s}\approx T_{s}\psi_{0}^{*}TX\oplus
T_{q}\mathbb{C}.$

In the splitting (8.5) the differential at $(t,\xi_{0},q)$ of
$\Phi_{1}|O_{t,s}$ is $\exp_{*}\theta(t,s,q)\oplus 0$, cf. (8.3), (8.4), and
of $\Phi_{2}|O_{t,s}$ it is $0\oplus\text{id}_{T_{q}\mathbb{C}}$. Since $\exp$
is biholomorphic on any $T_{x}X$, it follows that ${(\Phi|O_{t,s})}_{*}$ is
invertible at all points of $[a,b]\times S\times I$, hence also in a
neighborhood in $[a,b]\times V$. As $\Phi$ is injective on $\\{(t,s)\\}\times
I$, a simple indirect argument shows that if

$O\subset[a,b]\times V=[a,b]\times(\psi_{0}^{*}TX)\times\mathbb{C}$

is sufficiently shrunk, all $\Phi|O_{t,s}$ will be biholomorphic.

We can take $O$ of form $[a,b]\times O^{\prime}\times\Omega$, with
$O^{\prime}\subset\psi_{0}^{*}TX$ a neighborhood of the zero section, and
$\Omega\subset\mathbb{C}$ a convex neighborhood of $I$. Let
$\Delta\subset\mathbb{C}$ be the unit disc and $f:\Delta\to\Omega$
biholomorphic, that maps $0$ to $0$. Choose a norm $|\,\,|_{1}$ on
$\psi_{0}^{*}TX$ whose unit ball bundle is contained in $O^{\prime}$, and
define a norm $|\,\,|$ on $V$ by

$|(\xi,\lambda)|=\max\big{(}|\xi|_{1},|\lambda|\big{)},\qquad\xi\in\psi_{0}^{*}TX,\quad\lambda\in\mathbb{C},$

whose unit ball bundle $U$ is contained in $O^{\prime}\times\Delta$. The
formula

(8.6) $\Psi(t,\xi,\lambda)=\Phi\big{(}t,\xi,f(\lambda)\big{)},\qquad
t\in[a,b],\quad(\xi,\lambda)\in U,$

then defines a continuous $\Psi:[a,b]\times U\to X\times\mathbb{C}$, whose
restrictions to $\\{t\\}\times U_{s}$ are biholomorphisms to open subsets of
$X\times\mathbb{C}$, as (a) claims.

On $[a,b]\times S\times I$, by (8.4), pr${}_{X}\circ\phi$ and
pr${}_{X}\circ\Phi$ agree with $\psi$. Therefore Lemma 8.4c implies that, if
$O$ above is taken a sufficiently small neighborhood of $[a,b]\times S\times
I$, then $d\big{(}\mathfrak{x}(s),\text{pr}_{X}(\Phi(a,v))\big{)}<\varepsilon$
for $v$ in the fiber $V_{s}\cap O$. Hence
$d\big{(}\mathfrak{x}(s),\text{pr}_{X}(\Psi(a,v))\big{)}<\varepsilon$ when
$v\in U_{s}$: (d) is also satisfied.

Finally, by (8.4) and by Lemma 8.4a,b (recall that $S\subset\psi_{0}^{*}TX$
and $S\times\\{0\\}\subset V$ are the zero sections)

$\Phi(t,s,q_{j})=\phi(t,s,q_{j})=\big{(}\psi(t,s,q_{j}),q_{j}\big{)}=\begin{cases}\big{(}\mathfrak{x}(s),0\big{)}&\text{
if }s\in S,\,j=0\\\ \big{(}\mathfrak{y}_{t}(s),q_{j}\big{)}&\text{ if }s\in
S_{j},\,j=1,\dots,J.\end{cases}$

Therefore if we define $\mathfrak{v}(s)=f^{-1}(q_{j})$ when $s\in S_{j}$,
$j=0,\dots J$, $\Psi$ will satisfy the remaining requirements (b,c) as well.

##  9 The extension theorem in mapping spaces

In this section we prove Theorem 7.5. Since the spaces
$C(S,X,A,\mathfrak{x}_{0})\subset B(S,X,A,\mathfrak{x}_{0})$ do not change if
$\mathfrak{x}_{0}$ is replaced by any other $\mathfrak{x}\in
C(S,X,A,\mathfrak{x}_{0})$, we will take $\mathfrak{x}=\mathfrak{x}_{0}$.
Recall the notion of basic neighborhoods
$D_{A}(\mathfrak{y})\subset\mathfrak{B}$, defined by a neighborhood $D\subset
X\times X$ of the diagonal, see (5.1). If $D$ is as in Lemma 2.1, then
$D_{A}(\mathfrak{y})$ is contractible.

Given the $E|\mathfrak{C}$ valued germ $\bf f$ of Theorem 7.5, let
$C\in\mathbb{R}$ be such that any analytic continuation ${\bf h}_{t}$ of $\bf
f$ along any path $\mathfrak{x}_{t}$ in $\mathfrak{C}$ satisfies $p\big{(}{\bf
h}_{t}(\mathfrak{x}_{t})\big{)}\leq C$. Lemmas 9.1, 9.3 below continue ${\bf
f}^{\text{ab}}$ along special paths $\mathfrak{y}_{t}$ in $\mathfrak{B}$. Fix
$D$ above so that $\bf f$ is the germ of a holomorphic section $f$ of
$E|\mathfrak{C}\cap D_{A}(\mathfrak{x})$ and $f$ has an Aron–Berner–type
extension $f^{\text{ab}}$ to a section of
$E^{\prime\prime}|D_{A}(\mathfrak{x})$. Then ${\bf f}^{\text{ab}}$ is the germ
of $f^{\text{ab}}$.

######  Lemma 9.1.

Consider a path $[0,1]\ni t\mapsto\mathfrak{y}_{t}\in\mathfrak{B}$ starting at
$\mathfrak{x}$. Assume $a\in(0,1)$ is such that $\mathfrak{y}_{t}\in
D_{A}(\mathfrak{x})$ when $t\leq a$. Assume also that there are a normed
topological vector bundle $\pi:(V,|\,\,|)\to S$ with unit ball bundle $U\to
S$, a $\mathfrak{v}\in B_{A}(V)$, $\sup_{S}|\mathfrak{v}|<1$, and a continuous
$\Psi:[a,1]\times U\to X$ such that for every $t\in[a,1]$ and $s\in S$
(a) $\Psi$ maps the fibers $\\{t\\}\times U_{s}$ biholomorphically on open
subsets of $X$;
(b) the zero vector $0_{s}\in V_{s}$ satisfies
$\Psi(t,0_{s})=\mathfrak{x}(s)$;
(c) $\Psi(t,\mathfrak{v})=\mathfrak{y}_{t}$;
(d) $\Psi(a,v)\in D^{\mathfrak{x}(s)}$ for $v\in U_{s}$, cf. (2.3).
Then ${\bf f}^{\text{ab}}$ has an analytic continuation ${\bf g}_{t}$ along
$\mathfrak{y}_{t}$, such that $p\big{(}{\bf
g}_{t}(\mathfrak{y}_{t})\big{)}\leq C$.

Let $\Gamma\subset B_{A}(V)$ be the unit ball.

######  Lemma 9.2.

For $t\in[a,1]$ the map

(9.1) $\Psi_{t}^{\circ}:\Gamma\ni\mathfrak{u}\mapsto\Psi(t,\mathfrak{u})\in
D_{A}(\mathfrak{x}),\quad\text{ cf. (d) above,}$

is biholomorphic on its image, an open subset of $D_{A}(\mathfrak{x})$.

###### Proof.

Fix $t\in[a,1]$. First we claim that the map

$G:U\ni v\mapsto\big{(}\pi v,\Psi(t,v)\big{)}\in S\times X$

is injective and open. Injectivity follows from (a) of Lemma 9.1, since $G$
maps $U_{s}$ to $\\{s\\}\times X$ injectively. That $G$ is open is a local
property. Accordingly, we can assume that $V\to S$ is trivial,
$V=S\times\mathbb{C}^{r}$. Suppose $\Psi(t,s_{0},u_{0})=z_{0}\in X$ with some
$(s_{0},u_{0})\in U$, and let $B\subset\mathbb{C}^{r}$ be a small compact
neighborhood of $u_{0}$. The topological degree
$\deg\big{(}\Psi(t,s_{0},\cdot),B,z_{0}\big{)}=1$. Hence
$\deg\big{(}\Psi(t,s,\cdot),B,z\big{)}=1$ for $s,z$ close to $s_{0},z_{0}$;
$\Psi(t,s,\cdot)$ attains the value $z$ in $B$, and it follows that $G$ is
indeed open. This implies that $\Psi_{t}^{\circ}$ is open. It is also
injective.

The inverse $\Phi$ of $G$ is a homeomorphism between an open $O\subset S\times
X$ and $U$, holomorphic on $O\cap\big{(}\\{s\\}\times X)$, and satisfies
$\Psi\big{(}t,\Phi(s,z)\big{)}=z$, $s\in S$, $z\in D^{\mathfrak{x}(s)}$. The
map

(9.2)
$\Phi^{\circ}:\Psi^{\circ}_{t}(\Gamma)\ni\mathfrak{z}\mapsto\Phi\circ(\text{id}_{S}\times\mathfrak{z})\in\Gamma$

is the inverse of $\Psi_{t}^{\circ}$, and both $\Phi^{\circ},\Psi_{t}^{\circ}$
are holomorphic. To check the holomorphy of $\Psi_{t}$, say, take a local
coordinate $\psi_{\mathfrak{z}}$ on $\mathfrak{B}$ as in (5.2) with $F:D\to
TX$ of Lemma 2.1,

$\psi_{\mathfrak{z}}:D_{A}(\mathfrak{z})\ni\mathfrak{w}\mapsto
F\circ(\mathfrak{z}\times\mathfrak{w})\in B_{A}\big{(}\mathfrak{z}^{*}TX).$

The formula $H(v)=F\big{(}\mathfrak{z}(\pi v),\Psi(t,v)\big{)}\in
T_{\mathfrak{z}(\pi v)}X$ defines a map from a Borel subset of $V$ to
$\mathfrak{z}^{*}TX$ that satisfies the assumptions of Lemma 1.3. Therefore
$\psi_{\mathfrak{z}}\circ\Psi^{\circ}_{t}=F\circ(\mathfrak{z}\times\Psi_{t}^{\circ}),$
the restriction of $H^{\circ}$ of that lemma, is holomorphic on the open
subset of $\Gamma$ where it is defined. This means $\Psi_{t}^{\circ}$ is
indeed holomorphic. A similar argument proves for $\Phi^{\circ}$; therefore
both are biholomophic.

###### Proof of Lemma 9.1.

For values $t\leq a$ the germs of $f^{\text{ab}}$ at $\mathfrak{y}_{t}$ supply
the continuation ${\bf g}_{t}$. To continue ${\bf g}_{a}$ to values $t\geq a$,
we will prove that with $\Psi^{\circ}_{t}$ of (9.1):
(1) ${\bf f}^{\text{ab}}$ is the germ of a holomorphic section $g_{t}$ of
$E^{\prime\prime}|\Psi_{t}^{\circ}(\Gamma)$, $\sup p^{\prime\prime}(g_{t})\leq
C$;
(2) the germ of $g_{a}$ at $\mathfrak{y}_{a}$ is ${\bf g}_{a}$;
(3) the germs ${\bf g}_{t}$ of $g_{t}$ at $\mathfrak{y}_{t}$, $t\in[a,1]$,
provide an analytic continuation of ${\bf g}_{a}$.

(1) Fix $t\in[a,1]$. The pullback
$\Psi_{t}^{\circ*}(E,p)=(\bar{E},\bar{p})\to\Gamma$ is a holomorphic Banach
bundle with a flat norm $\bar{p}$. By Theorem 6.3 it is therefore trivial, and
its (bounded) holomorphic sections can be viewed as (bounded) holomorphic
functions with values in a fixed Banach space $(\mathfrak{Y},||\,\,||)$. Since
the germ ${\bf k}={\bf f}\circ\Psi_{t}^{\circ}$ has bounded analytic
continuation along any path in $\Gamma\cap\mathfrak{C}$ (supplied by the
pullback along of $\Psi_{t}^{\circ}$ of the corresponding continuation of $\bf
f$), it is the germ of a bounded holomorphic
$k:\Gamma\cap\mathfrak{C}\to\mathfrak{Y}$. This $k$ has an Aron–Berner–type
extension $k^{\text{ab}}$ to $\Gamma$ with $\sup
p^{\prime\prime}(k^{\text{ab}})\leq C$, constructed in section 4. Corollary
4.6 implies that the germ of $k^{\text{ab}}$ at $0$ is ${\bf
f}^{\text{ab}}\circ\Psi_{t}^{\circ}$, and this means that
$g_{t}=k^{\text{ab}}\circ\Phi^{\circ}$ will do, cf. (9.2).

(2) Since $\Psi_{a}^{\circ}(\Gamma)\subset D_{A}(\mathfrak{x})$ is connected,
and $f^{\text{ab}}$, $g_{a}$ share the same germ at $\mathfrak{x}$, namely
${\bf f}^{\text{ab}}$, on $\Psi^{\circ}_{a}(\Gamma)$ we have
$f^{\text{ab}}=g_{a}$. In particular, their germs at
$\mathfrak{y}_{a}=\Psi^{\circ}_{a}(\mathfrak{v})$ agree.

(3) Given $t\in[a,1]$, connect $\mathfrak{x}$ with $\mathfrak{y}_{t}$ by a
path in $\Psi_{t}^{\circ}(\Gamma)$. By continuity of $\Psi$, for
$\tau\in[a,1]$ close to $t$, this path is contained in
$\Psi_{\tau}^{\circ}(\Gamma)$. Since the germs of $g_{t},g_{\tau}$ at
$\mathfrak{x}$ agree, $g_{t},g_{\tau}$ agree in a neighborhood of the path,
hence in a neighborhood of $\mathfrak{y}_{t}$. But this means that the germs
${\bf g}_{t}$ depend continuously on $t$, and they continue ${\bf g}_{a}$
analytically. This completes the proof.

Again we metrize $X$ with a metric $d$, then $d_{S}$ of (2.2) metrizes
$\mathfrak{B}$.

######  Lemma 9.3.

There is a $\delta>0$ with the following property. Let
$S=\coprod_{j=0}^{J}S_{j}$ be a partition in Borel sets, $S_{0}=A$. Suppose
that $[0,1]\ni t\mapsto\mathfrak{y}_{t}\in\mathfrak{B}$ is a path starting at
$\mathfrak{x}$, and $a\in(0,1)$ is such that
(a) $d_{S}(\mathfrak{x},\mathfrak{y}_{t})<\delta$ when $t\in[0,a]$;
(b) $\mathfrak{y}_{t}|S_{j}$ is constant when $t\in[a,1]$, $j=1,2,\dots,J$.
Then ${\bf f}^{\text{ab}}$ has an analytic continuation ${\bf g}_{t}$ along
$\mathfrak{y}_{t}$, such that $p\big{(}{\bf
g}_{t}(\mathfrak{y}_{t})\big{)}\leq C$.

###### Proof.

With $X^{+}=X\times\mathbb{C}$ and pr${}_{X}:X^{+}=X\times\mathbb{C}\to X$ the
projection, let $D^{+}=(\text{pr}_{X}\times\text{pr}_{X})^{-1}D\subset
X^{+}\times X^{+}$, a neighborhood of the diagonal,
$\mathfrak{x}^{+}=\mathfrak{x}\times 0\in C(S,X^{+})$,

$\mathfrak{B}^{+}=\\{\mathfrak{z}\in
B(S,X^{+}):\mathfrak{z}|A=\mathfrak{x}^{+}|A\\},$

and $\mathfrak{C}^{+}$ the component of $\mathfrak{x}^{+}$ in
$\mathfrak{B}^{+}\cap C(S,X^{+})$. Choose $\varepsilon>0$ so that
$D(\mathfrak{x})$ contains the $d_{S}$ metric ball of radius $\varepsilon$
about $\mathfrak{x}$. Given $\mathfrak{x}$ and $\varepsilon$, the $\delta>0$
provided by Lemma 8.1 will do.

Indeed, if (a), (b) of our lemma hold, choose a normed vector bundle
$(V,|\,\,|)\to S$ with unit ball bundle $U\subset V$, $\mathfrak{v}$ in the
unit ball $\Gamma\subset B_{A}(V)$, and a map $\Psi:[a,1]\times U\to X^{+}$ as
guaranteed by Lemma 8.1. Consider the projection
$P:\mathfrak{B}^{+}\to\mathfrak{B}$ given by
$P(\mathfrak{z})=\text{pr}_{X}\circ\mathfrak{z}$; it is holomorphic and maps
$\mathfrak{C}^{+}$ to $\mathfrak{C}$. The pullback $f\circ P$ is a holomorphic
section of $P^{*}E|\big{(}\mathfrak{C}^{+}\cap
D^{+}(\mathfrak{x}^{+})\big{)}$, whose germ at $\mathfrak{x}^{+}$ we denote
$\bf h$. Now $\bf h$ analytically continues along any path $r$ in
$\mathfrak{C}^{+}$, the continuations provided by the lifts of the
continuations along $P\circ r$; these continuations are bounded, with the same
bound $C$ as for $\bf f$. Let us set
$\mathfrak{z}_{t}=\Psi(t,\mathfrak{v})\in\mathfrak{B}^{+}$, $t\in[0,1]$. We
are in the situation of Lemma 9.1, with $X$ replaced by $X^{+}$,
$\mathfrak{B}$ by $\mathfrak{B}^{+}$, $\mathfrak{y}_{t}$ by
$\mathfrak{z}_{t}$. By the lemma ${\bf h}^{\text{ab}}$ has an analytic
continuation ${\bf k}_{t}$ along $\mathfrak{z}_{t}$. It remains to show that
the $P^{*}E^{\prime\prime}$ valued germs ${\bf k}_{t}$ are pullbacks of germs
${\bf g}_{t}$ that form the analytic continuation of ${\bf f}^{\text{ab}}$
along $\mathfrak{y}_{t}$.

Let us call a holomorphic vector field $\xi$ on $\mathfrak{B}^{+}$ vertical if
it is tangential to the fibers $P^{-1}(\mathfrak{y})$,
$\mathfrak{y}\in\mathfrak{B}$. Consider a Banach space $\mathfrak{Y}$ and a
$\mathfrak{Y}$ valued holomorphic germ $\bf k$ at some
$\mathfrak{z}\in\mathfrak{B}^{+}$. It is of form ${\bf k}={\bf g}\circ P$ with
a $\mathfrak{Y}$ valued holomorphic germ $\bf g$ at $P(\mathfrak{z})$ if and
only if the derivatives $\xi{\bf k}=0$ for all vertical vector fields $\xi$.
Similarly, if $\bf k$ is a $P^{*}E^{\prime\prime}$ valued germ at
$\mathfrak{z}\in\mathfrak{B}^{+}$, we can compute its vertical derivatives
$\xi\bf k$, since the restriction of (a representative of) $\bf k$ to a fiber
$P^{-1}(\mathfrak{u})$ takes values in the fixed Banach space
$E^{\prime\prime}_{P(\mathfrak{u})}$; and $\bf k$ is a pullback if and only if
$\xi{\bf k}=0$ for all vertical $\xi$.

For the analytic continuation ${\bf k}_{t}$ of ${\bf h}^{\text{ab}}={\bf
k}_{0}$ constructed above, $\xi{\bf k}_{0}=0$, because by Corollary 4.6 ${\bf
k}_{0}$ is the pullback of ${\bf f}^{\text{ab}}$ along $P$. Hence its analytic
continuation $\xi{\bf k}_{t}$ also vanishes, and ${\bf k}_{t}$ is the pullback
of some germ ${\bf g}_{t}$ at $\mathfrak{y}_{t}$. By Lemma 7.2 ${\bf g}_{t}$
depend continuously on $t$, hence supply the analytic continuation of ${\bf
f}^{\text{ab}}$.

We will reduce continuation along arbitrary paths to continuation along paths
as in Lemma 9.3 in the following way. By a partition we will mean a
decomposition $S=\coprod_{j=0}^{J}S_{j}$ into Borel sets such that $S_{0}=A$,
$S_{j}\neq\emptyset$, $j=1,\dots,J$. Given such a partition, denoted $\Pi$,
let

(9.3)
$\mathfrak{B}_{\Pi}=\\{\mathfrak{y}\in\mathfrak{B}:\mathfrak{y}|S_{j}=\text{const},\text{
for }j=1,\dots,J\\}.$

If $\Pi^{\prime}$ refines $\Pi$ (notation: $\Pi^{\prime}\succ\Pi$), then
$\mathfrak{B}_{\Pi^{\prime}}\supset\mathfrak{B}_{\Pi}$.

######  Lemma 9.4.

(a) $\mathfrak{B}_{\Pi}\subset\mathfrak{B}$ is a closed submanifold,
biholomorphic to $X^{J}$;

(b) If $\Pi_{0}$ is a partition, then
$\bigcup_{\Pi\succ\Pi_{0}}\mathfrak{B}_{\Pi}$ is dense in $\mathfrak{B}$;

(c) Let $c\in\mathbb{R}$, $\mathfrak{U}\subset\mathfrak{B}$ open,
$K\to\mathfrak{U}$ a holomorphic Banach bundle with a flat norm $p$, and
$\Pi_{0}$ a partition. Suppose for each $\Pi\succ\Pi_{0}$ we are given a
holomorphic section $h_{\Pi}$ of $K|\mathfrak{B}_{\Pi}\cap\mathfrak{U}$ such
that $p(h_{\Pi})\leq c$, and $\Pi^{\prime}\succ\Pi$ implies
$h_{\Pi^{\prime}}=h_{\Pi}$ on $\mathfrak{B}_{\Pi}\cap\mathfrak{U}$. Then there
is a unique holomorphic section $h$ of $K|\mathfrak{U}$ such that
$h|\mathfrak{B}_{\Pi}\cap\mathfrak{U}=h_{\Pi}$.

(d) Let $K\to\mathfrak{B}$ be a holomorphic Banach bundle with a flat norm. If
a $K$ valued holomorphic germ $\bf h$ at some
$\mathfrak{z}\in\mathfrak{B}_{\Pi_{0}}$ has analytic continuation along any
path in $\mathfrak{B}_{\Pi}$ for $\Pi\succ\Pi_{0}$, and these continuations
have the same bound $c$, then $\bf h$ has an analytic continuation along any
path in $\mathfrak{B}$, bounded by $c$.

###### Proof.

(a) $\mathfrak{B}_{\Pi}$ is clearly a closed subset. If
$\mathfrak{y}\in\mathfrak{B}_{\Pi}$, local coordinates $\psi_{\mathfrak{y}}$
as in (5.2) send the neighborhood $D_{A}(\mathfrak{y})$ to an open
$\mathfrak{V}\subset B_{A}(\mathfrak{y}^{*}TX)$. For all $j\geq 1$ the fibers
of $\mathfrak{y}^{*}TX$ over $S_{j}$ are the same tangent spaces to $X$, hence
one can talk about sections of $\mathfrak{y}^{*}TX$ being constant over
$S_{j}$. Those bounded Borel sections of $\mathfrak{y}^{*}TX$ that vanish on
$S_{0}$ and are constant over each $S_{j}$, $j\geq 1$, form a finite
dimensional subspace

$B_{\Pi}(\mathfrak{y}^{*}TX)\subset B_{A}(\mathfrak{y}^{*}TX),$

and the local coordinate $\psi_{\mathfrak{y}}$ sends $\mathfrak{B}_{\Pi}\cap
D_{A}(\mathfrak{y})$ to $B_{\Pi}(\mathfrak{y}^{*}TX)\cap\mathfrak{V}$.
Therefore $\mathfrak{B}_{\Pi}$ is indeed a submanifold of $\mathfrak{B}$. One
can then check that the map that sends $(x_{1},\dots,x_{J})\in X^{J}$ to
$\mathfrak{z}\in\mathfrak{B}_{\Pi}$ defined by $\mathfrak{z}(s)=x_{j}$ if
$s\in S_{j}$, $j=1,\dots,J$, is biholomorphic.

(b) Let $\mathfrak{y}\in\mathfrak{B}$ and $\varepsilon>0$. To find a partition
$\Pi\succ\Pi_{0}$ and $\mathfrak{z}\in\mathfrak{B}_{\Pi}$ in the $\varepsilon$
neighborhood of $\mathfrak{y}$, cover the closure of $\mathfrak{y}(S)\subset
X$ by finitely many balls of radius $\varepsilon$, choose $\Pi\succ\Pi_{0}$,
$S=\coprod_{j=0}^{J}S_{j}$, so that each $\mathfrak{y}(S_{j})$ is contained in
one of those balls, centered at, say, $y_{j}\in X$, $j\geq 1$. Then
$\mathfrak{z}\in\mathfrak{B}_{\Pi}$ given by

$\mathfrak{z}(s)=\begin{cases}\mathfrak{x}(s)&\text{ when }s\in S_{0}=A\\\
y_{j}&\text{ when }s\in S_{j},\quad i=1,\dots,J\end{cases}$

will do.

(c) We can assume that $\mathfrak{U}$ is contained in a coordinate
neighborhood about some $\mathfrak{y}\in\mathfrak{B}$ and that $(K,p)$ is
trivial over it. Then instead of sections one can talk about Banach space
valued functions. The local coordinate $\psi_{\mathfrak{y}}$ maps
$\mathfrak{U}$ to a neighborhood $\mathfrak{W}\subset
B_{A}(\mathfrak{y}^{*}TX)$ of the zero section, and
$\mathfrak{B}_{\Pi}\cap\mathfrak{U}$ into $B_{\Pi}(\mathfrak{y}^{*}TX)$ from
the proof of part (a). A hermitian metric on $X$ induces a norm on
$B_{A}(\mathfrak{y}^{*}TX)$, and we can assume that $\mathfrak{W}$ is the unit
ball.

Accordingly, the claim will be proved once we show the following. Let
$\mathfrak{X}$, $(\mathfrak{Y},||\,\,||)$ be Banach spaces,
$\mathfrak{W}\subset\mathfrak{X}$ the unit ball, and $c\in\mathbb{R}$. Suppose
for each partition $\Pi\succ\Pi_{0}$ we are given a closed subspace
$\mathfrak{X}_{\Pi}\subset\mathfrak{X}$ and $h_{\Pi}\in{\cal
O}(\mathfrak{X}_{\Pi}\cap\mathfrak{W},\mathfrak{Y})$ with the following
properties. First, $||h_{\Pi}||\leq c$; second,
$\mathfrak{X}_{\Pi^{\prime}}\supset\mathfrak{X}_{\Pi}$ and
$h_{\Pi^{\prime}}|\mathfrak{X}_{\Pi}\cap\mathfrak{W}=h_{\Pi}$ if
$\Pi^{\prime}\succ\Pi$; and finally,
$\mathfrak{Z}=\bigcup_{\Pi\succ\Pi_{0}}\mathfrak{X}_{\Pi}$ is dense in
$\mathfrak{X}$. Then there is a unique $h\in{\cal
O}(\mathfrak{W},\mathfrak{Y})$ such that
$h|\mathfrak{X}_{\Pi}\cap\mathfrak{W}=h_{\Pi}$ and $||h||\leq c$. (Of course,
partitions here are irrelevant, $\Pi$ could denote elements of an arbitrary
directed set.)

The density of $\mathfrak{Z}$ implies that $h$ is unique. To construct $h$,
define $g:\mathfrak{Z}\cap\mathfrak{W}\to\mathfrak{Y}$ by $g(x)=h_{\Pi}(x)$ if
$x\in\mathfrak{X}_{\Pi}\cap\mathfrak{W}$. Cauchy estimates for the derivatives
of $h_{\Pi}$ imply that on a concentric ball
$\mathfrak{W}_{r}\subset\mathfrak{W}$ of radius $r$ the Lipschitz constant of
$h_{\Pi}$ is $\leq c/(1-r)$, hence the same holds for $g$. In particular, if
$x\in\mathfrak{W}$ and a sequence $x_{i}\in\mathfrak{Z}\cap\mathfrak{W}$
converges to $x$, then $g(x_{i})$ form a Cauchy sequence. Define
$h(x)=\lim_{i}g(x_{i})$; the limit is independent of the sequence $x_{i}\to
x$. The function $h$ constructed is continuous and its norm is bounded by $c$.
To complete the proof we need to check that for any $x,y\in\mathfrak{X}$ and
$\Lambda=\\{\lambda\in\mathbb{C}:x+\lambda y\in\mathfrak{W}\\}$, the function
$f:\Lambda\ni\lambda\mapsto h(x+\lambda y)$ is holomorphic. Let
$\Lambda_{0}\subset\Lambda$ be open and relatively compact. Choose sequences
$\Pi_{i}\succ\Pi_{0}$, and $x_{i},y_{i}\in\mathfrak{X}_{\Pi_{i}}$ convergent
to $x,y$. Then $f_{i}(\lambda)=h(x_{i}+\lambda
y_{i})=h_{\Pi_{i}}(x_{i}+\lambda y_{i})$ is holomorphic on $\Lambda_{0}$ when
$i$ is sufficiently large. Since $||f_{i}||\leq c$ and $f_{i}\to f$ on
$\Lambda_{0}$, it follows that $f$ is holomorphic, and therefore so is $h$.

(d) Let $r:[a,b]\to\mathfrak{B}$ be a path, $\bf h$ a germ at
$r(a)=\mathfrak{z}$. Let $\mathfrak{W}_{t}\subset D_{A}\big{(}r(t)\big{)}$ be
neighborhoods of $r(t)$ whose image under the local coordinate $\psi_{r(t)}$
is convex, $r(a)\in\mathfrak{W}_{t}$ for $t$ in a neighborhood of $a$, and
$\bf h$ is the germ of a holomorphic section $h^{0}$ of $K|\mathfrak{W}_{a}$.
Then $\mathfrak{B}_{\Pi}\cap\mathfrak{W}_{t}$ for each $\Pi$ is contractible
or empty. Choose $a=t_{0}<t_{1}<\dots<t_{k}=b$ so that
$r[t_{j-1},t_{j}]\subset\mathfrak{W}_{t_{j}}$ when $1\leq j\leq k$. By
recurrence we construct holomorphic sections $h^{j}$ of
$K|\mathfrak{W}_{t_{j}}$ that define the analytic continuation of $\bf h$
along $r$. If $j\geq 1$ and $h^{i}$ for $i<j$ has been constructed, choose a
partition $\Pi_{1}\succ\Pi_{0}$ and $\mathfrak{w}$ in the connected component
of $r(t_{j-1})$ in
$\mathfrak{B}_{\Pi_{1}}\cap\mathfrak{W}_{t_{j-1}}\cap\mathfrak{W}_{t_{j}}$.
The analytic continuations of the germ of $h^{j-1}$ at $\mathfrak{w}$ along
paths in $\mathfrak{B}_{\Pi}\cap\mathfrak{W}_{t_{j}}$, $\Pi\succ\Pi_{1}$,
define holomorphic sections $h_{\Pi}$ of
$K|\mathfrak{B}_{\Pi}\cap\mathfrak{W}_{t_{j}}$. Since these sections satisfy
the conditions in (c), they are restrictions of a holomorphic section $h^{j}$
of $K|\mathfrak{W}_{t_{j}}$. By the uniqueness part of (c) the germs of
$h^{j-1}$ and $h^{j}$ agree at $\mathfrak{w}$, therefore in the connected
component of $\mathfrak{w}$ in
$\mathfrak{W}_{t_{j-1}}\cap\mathfrak{W}_{t_{j}}$. In particular, $h^{j-1}$ and
$h^{j}$ agree near $r(t_{j-1})$. Thus the collection $h^{j}$ indeed gives rise
to an analytic continuation of $\bf h$ along $r$.

###### Proof of Theorem 7.5.

It suffices to construct continuations along paths $[0,1]\ni
t\mapsto\mathfrak{x}_{t}\in\mathfrak{B}$, $\mathfrak{x}_{0}=\mathfrak{x}$. Let
$\delta>0$ as in Lemma 9.3 and $a\in(0,1/2)$ be such that the
$\delta$–neighborhood $\mathfrak{U}\subset\mathfrak{B}$ of $\mathfrak{x}$ is
contained in $D_{A}(\mathfrak{x})$ and contains $\mathfrak{x}_{t}$ when
$t\in[0,2a]$. Choose a partition $S=\coprod_{i=0}^{I}T_{i}$, denoted
$\Pi_{0}$, such that $\mathfrak{B}_{\Pi_{0}}$ intersects $\mathfrak{U}$ (Lemma
9.4b), say, $\mathfrak{y}\in\mathfrak{B}_{\Pi_{0}}\cap\mathfrak{U}$. Since the
image of $\mathfrak{U}$ under the local coordinate $\psi_{\mathfrak{x}}$ is
convex, there is a path $[0,2a]\ni t\mapsto\mathfrak{y}_{t}\in\mathfrak{U}$
from $\mathfrak{x}$ to $\mathfrak{x}_{2a}$ such that
$\mathfrak{y}_{a}=\mathfrak{y}$; extend it to $2a\leq t\leq 1$ by
$\mathfrak{y}_{t}=\mathfrak{x}_{t}$. The germs of $f^{\text{ab}}$ provide
continuation of ${\bf f}^{\text{ab}}$ along both
$\mathfrak{x}_{t},\mathfrak{y}_{t}$, $t\in[0,2a]$. These continuations give
the same result at $\mathfrak{x}_{2a}$, that we denote ${\bf g}$. Let $\bf h$
be the germ of $f^{\text{ab}}$ at $\mathfrak{y}_{a}$. To produce analytic
continuation of ${\bf f}^{\text{ab}}$ along $\mathfrak{x}_{t}$, $t\in[0,1]$,
it suffices to show continuation of ${\bf g}$ along $\mathfrak{x}_{t}$,
$t\in[2a,1]$, or of $\bf h$ along $\mathfrak{y}_{t}$, $t\in[a,1]$. But Lemma
9.3 guarantees that this latter continuation exists along paths
$\mathfrak{y}^{\prime}_{t}$ for which
$\mathfrak{y}^{\prime}_{t}\in\mathfrak{B}_{\Pi}$ when $t\in[a,1]$; here
$\Pi\succ\Pi_{0}$ is arbitrary. Hence, by Lemma 9.4d, the continuation exists
along $\mathfrak{y}_{t}$ as well. This completes the proof.

##  10 The Monodromy theorem

In this section we prove the following generalization of Theorem 0.2. Consider
mapping spaces $\mathfrak{B},\mathfrak{C}$ determined by a compact Hausdorff
space $S$, a complex manifold $X$, a continuous $\mathfrak{x}\in C(S,X)$ and a
closed $A\subset S$, as before.

######  Theorem 10.1.

Suppose $X$ is simply connected, $E\to\mathfrak{B}$ is a holomorphic Banach
bundle with a flat norm $p$, and an $E|\mathfrak{C}$ valued holomorphic germ
$\bf f$ admits bounded analytic continuation along any path in $\mathfrak{C}$.
Then $\bf f$ is the germ of a holomorphic section $f$ of $E|\mathfrak{C}$. If,
furthermore, $X$ is compact, then $f$ is a horizontal section (Definition
6.4).

Fix a partition $S=\coprod_{j=0}^{J}S_{j}$ as in section 9, denote it
$\Pi_{0}$, and let $\bar{\mathfrak{x}}\in\mathfrak{B}_{\Pi_{0}}$, see (9.3).
Given $\nu=0,1,\dots$, inclusions induce a directed system of homotopy
groups/sets and homomorphisms

(10.1)
$\pi_{\nu}(\mathfrak{B}_{\Pi},\bar{\mathfrak{x}})\to\pi_{\nu}(\mathfrak{B}_{P},\bar{\mathfrak{x}}),\qquad
P\succ\Pi\succ\Pi_{0}.$

######  Lemma 10.2.

$\pi_{\nu}(\mathfrak{B},\bar{\mathfrak{x}})$ is isomorphic to the direct limit
of the system (10.1). Hence, if $X$ is connected (our standing assumption),
resp. simply connected, then so is $\mathfrak{B}$.

###### Proof.

We need to show that any continuous map
$(S^{\nu},*)\to(\mathfrak{B},\bar{\mathfrak{x}})$, and any homotopy between
two such maps, can be deformed to a map, respectively homotopy, into
$\mathfrak{B}_{\Pi}$ with some partition $\Pi\succ\Pi_{0}$; and that any
homotopy in $(\mathfrak{B},\bar{\mathfrak{x}})$ between continuous maps
$f_{1},f_{2}:(S^{\nu},*)\to(\mathfrak{B}_{\Pi},\bar{\mathfrak{x}})$ can be
deformed, fixing $f_{1},f_{2}$, to a homotopy in some
$(\mathfrak{B}_{P},\bar{\mathfrak{x}})$, $P\succ\Pi$. Once this done, we can
define a homomorphism
$\pi_{\nu}(\mathfrak{B},\bar{\mathfrak{x}})\to\varinjlim\pi_{\nu}(\mathfrak{B}_{\Pi},\bar{\mathfrak{x}})$
by first sending the class in $\pi_{\nu}(\mathfrak{B},\bar{\mathfrak{x}})$ of
an $f:(S^{\nu},*)\to(\mathfrak{B},\bar{\mathfrak{x}})$ to an
$f_{1}:(S^{\nu},*)\to(\mathfrak{B}_{\Pi},\bar{\mathfrak{x}})$ homotopic to
$f$, and then to the element of
$\varinjlim\pi_{\nu}(\mathfrak{B}_{P},\bar{\mathfrak{x}})$ that the class
$[f_{1}]\in\pi_{\nu}(\mathfrak{B}_{\Pi},\bar{\mathfrak{x}})$ induces. This
homomorphism is automatically bijective.

The deformation statements are special cases of the following: Let $T$ be a
compact Hausdorff space, $C\subset T$ closed. For any continuous
$\phi:(T,C)\to(\mathfrak{B},\mathfrak{B}_{\Pi})$ there is a continuous
deformation $\phi_{\tau}:(T,C)\to(\mathfrak{B},\mathfrak{B}_{\Pi})$,
$0\leq\tau\leq 1$, such that $\phi_{0}=\phi$, $\phi_{\tau}|C=\phi|C$ for all
$\tau$, and $\phi_{1}(T)\subset\mathfrak{B}_{P}$ with some $P\succ\Pi$. To
verify this, we will use $D\subset X\times X$, $D^{x}$, $F:D\to TX$ of Lemma
2.1, and $\exp:TX\to X$ of (2.4). Since $\phi(t)\in\mathfrak{B}$ is a
continuous function of $t\in T$, and each $\phi(t)(S)\subset X$ has compact
closure, there is a compact $L\subset X$ that contains all $\phi(t)(S)$. Cover
$L$ by finitely many coordinate neighborhoods $D^{x_{i}}$, and choose a
partition $P\succ\Pi$, given by $S=\coprod_{i=0}^{I}R_{i}$, such that
$\phi(t)(R_{i})\subset D^{x_{i}}$ for every $t\in T$ and $i=1,\dots,I$. If
$s_{i}\in R_{i}$, $i\geq 1$, then

$\phi_{\tau}(t)(s)=\begin{cases}\phi(t)(s)=\mathfrak{x}(s)&\text{ if }s\in
R_{0}=A\\\ \exp\big{\\{}(1-\tau)F\big{(}x_{i},\phi(t)(s)\big{)}+\tau
F\big{(}x_{i},\phi(t)(s_{i})\big{)}\big{\\}}&\text{ if }s\in R_{i},\quad
i=1,\dots,I\end{cases}$

is the deformation sought.

The second statement of the lemma follows because by Lemma 9.4a the spaces
$\mathfrak{B}_{\Pi}$ are homeomorphic to Cartesian powers of $X$, hence for
(simply) connected $X$ they themselves are (simply) connected.

###### Proof of Theorem 10.1.

Since $\mathfrak{B}$ is connected and simply connected by Lemma 10.2, $(E,p)$
is trivial. Again, instead of sections of $(E,p)$ we can talk about functions
with values in some fixed Banach space $(\mathfrak{Y},||\,\,||)$. Accordingly,
we take $\bf f$ to be a $\mathfrak{Y}$ valued germ. By Theorem 7.5 ${\bf
f}^{\text{ab}}$ admits bounded analytic continuation along any path in
$\mathfrak{B}$; again because $\mathfrak{B}$ is simply connected, this implies
that ${\bf f}^{\text{ab}}$ is the germ of a holomorphic
$g:\mathfrak{B}\to\mathfrak{Y}^{\prime\prime}$. Hence $\bf f$ is the germ of
the holomorphic function $f=g|\mathfrak{C}$. Since its germ $\bf f$ takes
values in $\mathfrak{Y}\subset\mathfrak{Y}^{\prime\prime}$, all the values
that $f$ takes are in $\mathfrak{Y}$.

If, in addition, $X$ is compact, then the submanifolds
$\mathfrak{B}_{\Pi}\approx X^{J}$ are also compact (and connected), hence the
restrictions $g|\mathfrak{B}_{\Pi}$ are constant. As
$\bigcup_{\Pi}\mathfrak{B}_{\Pi}$ is dense in $\mathfrak{B}$, Lemma 9.4b, this
implies that $g$ itself is constant, and therefore $f$ is constant.

##  11 Examples

In this section we have collected examples that show that some assumptions in
Theorems 7.5, 10.1 cannot be dispensed with, or that the conclusions cannot be
strengthened. We keep notation used in sections 7, 10.

######  Example 11.1.

There are mapping spaces $\mathfrak{C},\mathfrak{B}$ and a bounded holomorphic
function $f:\mathfrak{C}\to\mathbb{C}$ with germ $\bf f$ at some
$\mathfrak{x}_{0}$ such that ${\bf f}^{\text{ab}}$ is not the germ of a single
valued holomorphic function $g:\mathfrak{B}\to\mathbb{C}$.

Let $r\in(1,\infty)$, $X=\\{x\in\mathbb{C}:1/r<|x|<r\\}$ an annulus,
$S=[0,1]$, $A=\emptyset$, and $\mathfrak{x}_{0}\equiv 1$. Thus
$\mathfrak{B},\mathfrak{C}$ are open sets in the Banach spaces $B(S)$, $C(S)$.
If $\mathfrak{x}\in\mathfrak{C}$, define
$f(\mathfrak{x})=\sqrt{\mathfrak{x}(1)/\mathfrak{x}(0)}$, using the branch of
the square root obtained by continuation from $\sqrt{1}=1$ along the path
$[0,1]\ni t\mapsto\mathfrak{x}(t)/\mathfrak{x}(0)$. Let $\bf f$ be the germ of
$f$ at $\mathfrak{x}_{0}$. In a neighborhood of $\mathfrak{x}_{0}$, we can
write $f(\mathfrak{x})=\sqrt{\phi(\mathfrak{x})/\psi(\mathfrak{x})}$ with
holomorphic functions

(11.1)
$\phi(\mathfrak{x})=\mathfrak{x}(1),\quad\psi(\mathfrak{x})=\mathfrak{x}(0)$

on $C(S)$ and the principal branch of square root. Here $\phi,\psi$ are linear
functions, and their extensions $\phi^{\text{ab}},\psi^{\text{ab}}$ to $B(S)$
are given by the same formula (11.1). Since the Aron–Berner extension is
compatible with multiplication, it follows that ${\bf f}^{\text{ab}}$ is the
germ of

$g(\mathfrak{x})=\sqrt{\mathfrak{x}(1)/\mathfrak{x}(0)},\qquad\mathfrak{x}\in\mathfrak{B}\text{
in a neighborhood of }\mathfrak{x}_{0}.$

But ${\bf f}^{\text{ab}}$ has no single valued continuation e.g. along the
path $[0,1]\ni t\to\mathfrak{x}_{t}\in\mathfrak{B}$,

$\mathfrak{x}_{t}(s)=\begin{cases}e^{2\pi it}&\text{ if }s=1\\\ 1&\text{ if
}s\neq 1.\end{cases}$

######  Example 11.2.

There are mapping spaces $\mathfrak{B},\mathfrak{C}$ and (unbounded)
$f\in{\cal O}(\mathfrak{C})$ with germ $\bf f$ at some
$\mathfrak{x}_{0}\in\mathfrak{C}$ such that ${\bf f}^{\text{ab}}$ cannot be
continued along some path in $\mathfrak{B}$.

This is a minor reformulation of a result of Dineen. Let $X=\mathbb{C}$,
$S=\\{0,1/n:n\in\mathbb{N}\\}\subset\mathbb{R}$ with the topology inherited
from $\mathbb{R}$, $A=\emptyset$, and $\mathfrak{x}_{0}\equiv 0$. Thus
$\mathfrak{C}$ is biholomorphic to $c$, the space of convergent sequences of
complex numbers, and $\mathfrak{B}\supset\mathfrak{C}$ is biholomorphic to
$c^{\prime\prime}=l^{\infty}\supset c$. In particular, $\mathfrak{B}$ is
simply connected, whence if a germ $\bf g$ on it has analytic continuation
along any path, then it is the germ of a holomorphic function on
$\mathfrak{B}$. Since according to Dineen and Josefson [D71, J78] there are
holomorphic functions $f$ on $c$ that do not extend to $l^{\infty}$, our claim
follows. In fact, not only ${\bf f}^{\text{ab}}$ cannot be continued
analytically along some path, but no extension $\bf g$ of ${\bf f}$ to a germ
on $\mathfrak{B}$ has analytic continuation along all paths in $\mathfrak{B}$.

######  Example 11.3.

(a) Unless $X$ is contractible, it has a mapping space $\mathfrak{C}$ with a
flat normed holomorphic Banach bundle $(E,p)\to\mathfrak{C}$, and there is an
$E$ valued holomorphic germ at some $\mathfrak{x}_{0}\in\mathfrak{C}$ that has
bounded analytic continuation along any path in $\mathfrak{C}$, yet does not
extend to a holomorphic section of $E$. If $X$ is simply connected, then $E$
can be chosen a line bundle.

(b) If a Stein manifold $X$ is not contractible, it has a corresponding
mapping space $\mathfrak{C}$, and there is a $\mathbb{C}$ valued holomorphic
germ $\bf f$ at some $\mathfrak{x}_{0}\in\mathfrak{C}$ that continues
analytically along any path in $\mathfrak{C}$, yet $\bf f$ does not extend to
a holomorphic function $\mathfrak{C}\to\mathbb{C}$.

Both in (a) and (b) $X$ can be simply connected. Thus (b) shows that in
Theorem 10.1 the assumption that the analytic continuations are bounded cannot
be simply dropped, and (a) shows that even boundedness is not sufficient
unless it is known that $(E,p)$ extends to a flat bundle over
$\mathfrak{B}$.—If $X$ is contractible, then its mapping spaces are also
contractible, and the phenomena described in Example 11.3 will not occur.

If $X$ is not contractible, according to Whitehead it has a nontrivial
homotopy group $\pi_{\nu}(X,x_{0})$, $\nu\geq 1$ [W49, Theorem 1]. Let
$S=S^{\nu-1}$, $A\subset S$ a singleton, $\mathfrak{x}_{0}\equiv x_{0}$, and
$\mathfrak{C}=C(S,X,A,\mathfrak{x}_{0})$. Since
$\pi_{1}(\mathfrak{C},\mathfrak{x}_{0})\approx\pi_{\nu}(X,x_{0})\neq 0$, there
is a nontrivial covering $\pi:C\to\mathfrak{C}$ by a connected complex Banach
manifold $C$. For $\mathfrak{x}\in\mathfrak{C}$ let
$C_{\mathfrak{x}}=\pi^{-1}(\mathfrak{x})$, and let $E_{\mathfrak{x}}$ denote
the Hilbert space of functions $e:C_{\mathfrak{x}}\to\mathbb{C}$ with $l^{2}$
norm $p(e)=\big{(}\sum_{v\in C_{\mathfrak{x}}}|e(v)|^{2}\big{)}^{1/2}<\infty$.
On $E=\coprod_{\mathfrak{x}\in\mathfrak{C}}E_{\mathfrak{x}}$ the local
trivializations of $C\to\mathfrak{C}$ induce the structure of a holomorphic
Banach bundle over $\mathfrak{C}$, with flat norm $p$.

The pullback $\pi^{*}(E,p)\to C$ is trivial, and has a horizontal section $g$
(Definition 6.4) whose value at $y\in C_{\mathfrak{x}}$ is the delta function
$e\in E_{\mathfrak{x}}$ given by $e(y)=1$, $e(v)=0$ if $v\neq y$. The germ
$\bf g$ of $g$ at any point in $C_{\mathfrak{x}_{0}}$ pushes forward to a
holomorphic $E$ valued germ $\bf f$ at $\mathfrak{x}_{0}$; the pushforwards of
other germs of $g$ provide bounded analytic continuations of $\bf f$ along any
path in $\mathfrak{C}$; but $\bf f$ is not the germ of a section $f$ of $E$.
If it were, $g$ would be the pullback of $f$, and the singleton supports of
the functions $f(\mathfrak{x}):C_{\mathfrak{x}}\to\mathbb{C}$ would provide a
global section of the covering $C\to\mathfrak{C}$, which is impossible.

If $X$ is simply connected, then the above construction can be modified to
produce a line bundle $E\to\mathfrak{C}$. Namely, $\nu\geq 2$ then, whence
$G=\pi_{1}(\mathfrak{C},\mathfrak{x}_{0})\approx\pi_{\nu}(X,x_{0})$ is
Abelian. By Pontryagin duality it admits a nontrivial representation
$\rho:G\to\text{U}(1)$, [P66, Section 37]. This induces a flat line bundle
$(E,p)\to\mathfrak{C}$, obtained from the trivial line bundle
$\tilde{\mathfrak{C}}\times\mathbb{C}\to\tilde{\mathfrak{C}}$ over the
universal cover of $\mathfrak{C}$, endowed with the trivial norm = Hermitian
metric, by factoring out by the action

$\tilde{\mathfrak{C}}\times\mathbb{C}\ni(\mathfrak{y},\zeta)\mapsto\big{(}\gamma\mathfrak{y},\rho(\gamma)\zeta\big{)}\in\tilde{\mathfrak{C}}\times\mathbb{C},\qquad\gamma\in
G,$

($G$ acts on $\tilde{\mathfrak{C}}$ by deck transformations). As $\rho$ is
nontrivial, $(E,p)$, as a flat line bundle, is nontrivial, either—although, as
a holomorphic line bundle $E$ may very well be trivial. The section
$\tilde{\mathfrak{C}}\ni y\mapsto(y,1)\in\tilde{\mathfrak{C}}\times\mathbb{C}$
induces a multivalued horizontal section of $(E,p)$, any of whose germs at
$\mathfrak{x}_{0}$ is the $\bf f$ sought.

A further modification is possible when $X$ is Stein. In this case the
universal cover $\tilde{\mathfrak{C}}$ can be embedded in a Banach space. To
see this, we first embed $X$ holomorphically and properly into some Euclidean
space $\mathbb{C}^{N}$, and arrange that in the embedding $x_{0}$ is sent to
$0$. This exhibits $\mathfrak{C}$ as a closed complex direct submanifold222A
subset $M$ of a complex Banach manifold $N$ is a direct submanifold if for
each $q\in N$ there are a neighborhood $U\subset N$ of $q$, a complemented
subspace $A\subset B$ in a Banach space $B$, and an open $V\subset B$ such
that the pairs $(U,U\cap M)$ and $(V,V\cap A)$ are biholomorphic. of the
Banach space $C_{A}(S)^{\oplus N}$. The latter space is isomorphic to a finite
codimensional subspace of $C(S)^{\oplus N}\approx C[0,1]^{\oplus N}$, hence
has a Schauder basis, according to Miljutin and Schauder [M66, S27]. All this
implies that $\tilde{\mathfrak{C}}$ can be embedded in a Banach space as a
closed direct manifold. This is proved in [L10, Theorem 3.7], except the
assumption there is that $\mathfrak{C}$ should be embedded in a Banach space
with unconditional basis. However, the unconditionality of the basis there was
used only to be able to quote certain earlier results of cohomological nature
by Patyi and by this author. All those results follow from cohomology
vanishing in [LP07, Theorem 9.1], one of whose assumptions (“plurisubharmonic
domination”) could at the time be proved only when unconditional bases
existed. In the meantime, however, Patyi extended plurisubharmonic domination
to spaces with a Schauder basis [P11, P15]. This means that embeddibility of
the universal (or any) cover, [L10, Theorem 3.7], holds under the assumption
of Schauder, rather than unconditional, basis. This takes care of the
embedding of our $\tilde{\mathfrak{C}}$.

Once the embedding granted, for any pair of distinct points
$a,b\in\tilde{\mathfrak{C}}$ there is a $g\in{\cal O}(\tilde{\mathfrak{C}})$
that takes different values at $a,b$. If we choose $a,b$ both lie above
$\mathfrak{x}_{0}$, then the pushforward $\bf f$ to $\mathfrak{C}$ of the germ
of $g$ at $a$ continues analytically along any path in $\mathfrak{C}$, but
those continuations do not define a single valued function on $\mathfrak{C}$.

## References