# Valdivia’s lifting theorem for non-metrizable spaces. Preprint.

Thomas E. Gilsdorf

Department of Mathematics

Central Michigan University

Mt. Pleasant, MI 48859 USA

<EMAIL_ADDRESS>

Abstract. Valdivia’s lifting theorem of (pre) compact sets and convergent
(respectively, Cauchy) sequences from a quasi-(LB) space to a metrizable,
strictly barrelled space is extended to a strictly larger collection of range
spaces. Specifically, we assume that the range space has a sequential web
structure and do not require that it be metrizable, nor strictly barrelled,
and the range space need not even be barrelled. Distinguishing examples are
provided that include natural constructions of range spaces connected with
applications, such as $\mathcal{D}_{\Gamma}^{\prime}$, the space of
distributions having their wavefront sets in a specified closed cone $\Gamma$.
The same and other examples could also serve as domain spaces for Valdivia’s
closed graph theorem, revealing a much wider collection of domain spaces that
can be used in that result.

2020 Mathematics Subject Classification: Primary 46A30; Secondary 54C05,
54C10, 46A08, 47A05, 46A03.

Keywords: Quasi-(LB) space, strictly barrelled, web, lifting theorem, closed
graph theorem.

##  1\. Introduction.

Nearly a decade after DeWilde’s [13] popular work on the closed graph theorem
led to webbed spaces being a common version of that theorem, Valdivia [36]
used webbed space structures to prove distinct closed graph and lifting
theorems for locally convex spaces that were defined as quasi - (LB) spaces.
Valdivia’s results seemingly attracted less attention than might have been
expected, perhaps as ‘yet another closed graph theorem’. In fact, Valdivia’s
methods and results of [36] still lead an active life in research. In
particular, the general structure of quasi- (LB) spaces is used quite
successfully in areas such as compact resolutions in topological groups [20],
descriptive topology [17], the Fréchet - Urysohn property in various contexts
[10], and $K$ analyticity in spaces of continuous functions, [31], to name a
few. In this paper, we appeal to W. Robertson’s [29] original definition of
webs that facilitates convergence of sequences in non-metrizable locally
convex spaces so as to extend the lifting theorem of [36] from metrizable
strictly barrelled locally convex spaces as the range, to those having
sequential webs of [23]. The collection of locally convex spaces with
sequential webs includes those that are metrizable, and also includes some
collections of locally convex spaces that are not necessarily metrizable, such
as any sequentially retractive locally convex inductive limit; c.f. [23].
Thus, for example, the space $\mathcal{D}$ of test functions from the
classical Schwartz distribution theory is allowable here. In addition, the
sequential web condition does not require that the spaces be strictly
barrelled or even barrelled. Examples of range spaces are provided, including
that of the space $\mathcal{D}_{\Gamma}^{\prime}$ of distributions having
their wevefront sets in a specified closed cone $\Gamma$, equipped with a
locally convex topology, (denoted as the normal topology, [11, p. 1350]). This
space, and its dual space, $\mathcal{E}_{\Lambda}^{\prime}$ have become
important in mathematical physics, due to their role in the formulation of
quantum field theory in curved spacetime [11, p.1345]. We observe that
Valdivia’s closed graph theorem is also applicable to domain spaces that need
not be strictly barrelled, and could be applied to locally convex spaces that
need not even be barrelled. Thus, Valdivia’s results from [36] are deeper and
more applicable than at first glance.

Topological vector spaces having completing webs are commonly used for proving
closed graph and open mapping theorems, [1], [3], [7], [8], [9], [12], [16],
[21], [22], [28], [29], [35], [36], [38], among others. In [36], strict webs,
originally defined as strict réseau absorbants (see [13]), are equipped with
an ordered structure and used to prove substantial generalizations of the
closed graph and lifting theorems of DeWilde [13] using techniques that
connect back to those of Banach’s original closed graph theorem proof, [2].
The approach taken in [36] is to assume that the linear function and strands
of a web satisfy properties for closed graph, open mapping, and lifting
theorems, rather than assume the domain and range spaces be of a particular
type of locally convex space. See [26, 9.1.44, p. 346] for another example of
this approach. By replacing metrizability with the presence of sequential
webs, Valdivia’s lifting theorem [36, 6.4, p. 162] can be generalized to
spaces that need be neither strictly barrelled nor metrizable.

Valdivia’s original Lifting theorem is stated here:

Valdivia’s Lifting theorem. ([36, 6.4, p. 162]). Suppose $E$ and $F$ are
Hausdorff locally convex spaces such that $F$ is a quasi-(LB) space and $E$ is
metrizable and strictly barrelled. Let $T$ be a linear map from subspace $H$
of $F$ onto $E$ for which the graph of $T$ is fast closed in $F\times E$.
Suppose $\\{A_{\alpha}:\alpha\in\mathbb{N}^{\mathbb{N}}\\}$ is a quasi-(LB)
representation in $F$. Define

$V_{m_{1}\cdots m_{k}}=\bigcup\\{T(A_{\alpha}\cap
H):\alpha\in\mathbb{N}^{\mathbb{N}},\alpha=(a_{n}),\,a_{n}=m_{n},n\in\underline{k}\\}.$

Assume there exists a sequence $(r_{k})$ of positive integers such that
$\overline{V_{r_{1}\cdots r_{k}}}$ is a neighborhood of the origin in $E$ for
$k=1,2,\cdots$. Then the following properties hold:

  1. (a)

If $(x_{n})$ is a sequence in $E$ which converges to the origin, there exists
a sequence $(u_{n})$ in $H$, fast convergent to the origin in $F$ and such
that $Tu_{n}=x_{n},n=1,2,\cdots$.

  2. (b)

If $A$ is a precompact set in $E$, then there exists a subset $B$ in $H$, fast
precompact in $F$ and such that $T(B)=A$.

  3. (c)

If $(y_{n})$ is a Cauchy sequence in $E$, there exists a sequence $(v_{n})$ in
$H$, fast convergent in $F$ and such that $Tv_{n}=y_{n},n=1,2,\cdots$.

We will abbreviate the above assumption that there exists a sequence $(r_{k})$
of positive integers such that $\overline{V_{r_{1}\cdots r_{k}}}$ is a
neighborhood of the origin in $E$ for $k=1,2,\cdots$ with the term strand-
neighborhood condition and replace the assumptions that $E$ is strictly
barrelled and metrizable with the less restrictive assumption that $E$ has a
sequential web. The main result of this paper is:

######  Theorem 1.

(Lifting Theorem). Suppose $E$ and $F$ are Hausdorff locally convex spaces
such that $F$ is a quasi-(LB) space and $E$ has a sequential web. Let $T$ be a
linear map from a fast closed linear subspace $H$ of $F$ onto $E$ for which
the graph of $T$ is closed in $H\times E$. If the images under $T$ of the
quasi- (LB) representation of $F$ restricted to $H$ satisfy the strand -
neighborhood condition, then:

  1. (a)

Every convergent sequence in $E$ is the image under $T$ of a sequence in $H$
that is fast convergent in $F$.

  2. (b)

If in $E$ each precompact set is contained in the absolutely convex hull of a
null sequence, then every precompact subset of $E$ is the image under $T$ of a
subset of $H$ that is fast precompact in $F$.

  3. (c)

Suppose that in addition $E$ is metrizable. Then every Cauchy sequence in $E$
is the image under $T$ of a sequence in $H$ that is fast convergent in $F$.

Theorem 1 generalizes [36, 6.4, p. 162] to range spaces $E$ that possess
sequential webs but need not be metrizable and need not be strictly barrelled,
as will be seen in the examples presented in section 4. The outline of this
paper is as follows: The next section will consist of basic definitions,
including those of webs, réseau, quasi (LB) spaces, strictly barrelled spaces,
sequential webs, as well as statements of relevant results that will be used.
Section three consists of the proof of Theorem 1 and a couple of consequences
thereof. The penultimate section consists of distinguishing examples and basic
properties of the spaces involved. Of note is that many natural examples of
such spaces turn out to be non-barrelled, and this implies potential
applications to such spaces. This paper concludes with a few indications
regarding potential further developments.

##  2\. Preliminaries.

###  2.1. General

Throughout this paper, $E=(E,\tau)$ denotes a Hausdorff locally convex
topological vector space, or space, over $\mathbb{K}=\mathbb{R}\mbox{ or
}\mathbb{C}$. A convex, balanced set is absolutely convex, which we will call
a disk. The expression $absconv(A)$ denotes the absolutely convex hull of a
set $A$. A system of zero neighborhoods in a space $E$ with topology $\tau$
will be denoted by $\mathcal{N}_{0,E}$. When distinction of topologies is
needed, we use the notation $\tau_{E}$ for a space $E$. The set
$\\{1,2,\cdots,k\\}$ is denoted by $\underline{k}$. The closure of a set $A$
in $(E,\tau)$ will be denoted by $\overline{A}$, and by
$\overline{A}^{\,\tau_{E}}$ whenever it is necessary to clarify the topology
involved. Likewise, the interior of a set $A$ is $A^{\circ}$, and
$A^{\circ\,\tau_{E}}$ to clarify. A linear subspace $H$ of a space $F$ will be
denoted by $H\leq F$. We make typical use of Banach disks and related ideas
described next, which incidentally, are originally due to Grothendieck [24]:

######  Definition 1.

  1. (1)

Given an absolutely convex subset $B$ of a space $E$, the linear span of $B$
is denoted by $E_{B}$, and we equip $E_{B}$ with the linear topology given by
the Minkowski functional of $B$, namely, for any $x\in E_{B}$,
$\mu_{B}(x)=\inf\\{t>0:x\in tB\\}$. If additionally the disk $B$ is bounded,
then with this topology $(E_{B},\mu_{B})$ is a normed space and we write
$(E_{B},||\cdot||_{B})$. It is easy to see that a base of zero neighborhoods
for $(E_{B},||\cdot||_{B})$ is given by $\\{n^{-1}B:n\in\mathbb{N}\\}$. The
boundedness of $B$ implies that the injection
$(E_{B},||\cdot||_{B})\hookrightarrow E$ is continuous. When this normed space
is complete, $B$ is called a Banach disk.

  2. (2)

A sequence $(y_{n})$ in $E$ is Mackey convergent to $y$ if converges to $y$ in
$(E_{B},||\cdot||_{B})$, for some closed bounded disk $B$, [26, 5.1.29, p.
158]. In particular, $(y_{n})$ is fast convergent, if the disk $B$ is a Banach
disk, [26, 6.1.20, p. 171]. Equivalently a sequence $(y_{n})$ in $E$ is Mackey
convergent to $y$ [26, 5.1.1, p. 151] if there exists a sequence $(d_{n})$ of
positive numbers such that $d_{n}\rightarrow\infty$ as $n\rightarrow\infty$,
and $d_{n}(y_{n}-y)\rightarrow 0$ in $E$. A space $E$ satisfies the Mackey
convergence condition, if any null sequence is Mackey null [26, 5.1.29, p.
158].

  3. (3)

A linear subspace $H$ of a space $F$ is fast closed if, given a Banach disk
$B$ of $F$, $H\cap F_{B}$ is a closed subspace of $F$, cf [36, p. 150] .

  4. (4)

A subset $A$ of $E$ is fast (pre) compact if for some Banach disk $B$, $A$ is
(pre) compact in $(E_{B},||\cdot||_{B})$.

######  Definition 2.

An inductive limit $E=ind_{n}E_{n}$ is sequentially retractive [26, 8.5.32, p.
295], if given any null sequence in $E$, there exists $m\in\mathbb{N}$ such
that the sequence is contained in and null in $E_{m}$.

######  Definition 3.

(See [15, Def 1.1, p. 52]). A projective limit of a sequence of strong duals
of Fréchet-Schwartz spaces (i.e., DFS spaces) is called a PLS - space.

So as to distinguish the results here from other concepts of “lifting”, we
include (c.f. [12, VI.3.5, p. 117]):

######  Definition 4.

Consider a linear map $T:F\rightarrow E$, for locally convex spaces $E$ and
$F$. Then $T$ is a lifting if either of the following hold:

  1. (1)

Every (pre-) compact subset of $E$ is the image under $T$ of a (pre-) compact
subset of $F$.

  2. (2)

Every (fast) (pre-) compact set in $E$ is the image under $T$ of a fast (pre-)
compact subset of $F$.

General background information on topological vector spaces can be found in
[26] or [30].

###  2.2. Webs and Réseaux webs (absorbents) and a comparison

DeWilde‘s réseaux absorbents are commonly denoted with W. Robertson’s original
terminology of webs [29]. The two definitions give rise to distinct
representations, and because we will use both constructions, a slightly closer
look at each is relevant. Both definitions follow.

######  Definition 5.

([29, p. 714]) A (Robertson) web $\mathcal{W}$ on a space $F$ is a countable
collection $\mathcal{W}$ of balanced sets, arranged in layers, where the first
layer is given by $\\{W_{n_{1}}:n_{1}\in\mathbb{N}\\}$ and subsequent layers,
denoted by $\\{W_{n_{1}n_{2}\cdots
n_{k}}:k,n_{1},n_{2},\cdots,n_{k}\in\mathbb{N}\\}$, that satisfy the following
properties:

  1. (1)

For each $k\in\mathbb{N}$,

$W_{n_{1}n_{2}\cdots n_{k+1}}+W_{n_{1}n_{2}\cdots n_{k+1}}\subset
W_{n_{1}n_{2}\cdots n_{k}};$

  2. (2)

$\bigcup\\{W_{n_{1}}:n_{1}\in\mathbb{N}\\}\mbox{ is absorbing in }F,$

  3. (3)

For each $k\in\mathbb{N}$,

$\bigcup\\{W_{n_{1}n_{2}\cdots n_{k+1}}:n_{k+1}\in\mathbb{N}\\}\mbox{ is
absorbing in }W_{n_{1}n_{2}\cdots n_{k}}.$

  4. (4)

Given $(n_{1},n_{2},\cdots)=(n_{k})\in\mathbb{N}^{\mathbb{N}}$, the sequence
$(W_{n_{1}},W_{n_{1}n_{2}},\cdots)$ of sets from $\mathcal{W}$ is called a
strand, and is denoted by $(W_{k})$, that is,

$(W_{k})_{k\in\mathbb{N}}=(W_{k})=(W_{n_{1}n_{2}\cdots n_{k}}).$

In particular, when all subsets of a web are absolutely convex then for any
strand $(W_{k})$ of $\mathcal{W}$, item (1) becomes:

(*) $(\forall k\in\mathbb{N})\;\;\;\;\;W_{k+1}\subset\frac{1}{2}W_{k}.$

######  Definition 6.

([13, p. 14], [12]) A réseaux (absorbents) web on a space $F$ is a countable
collection $\mathcal{R}$ of sets, denoted by $\\{A_{n_{1}n_{2}\cdots
n_{k}}:k,n_{1},n_{2},\cdots,n_{k}\in\mathbb{N}\\}$, satisfying the following
properties:

  1. (1)

$\bigcup\\{A_{n_{1}}:n_{1}\in\mathbb{N}\\}=F,$

  2. (2)

For each $k\in\mathbb{N}$,

$\bigcup\\{A_{n_{1}n_{2}\cdots
n_{k+1}}:n_{k+1}\in\mathbb{N}\\}=A_{n_{1}n_{2}\cdots n_{k}}.$

The following is the completeness property usually assumed for webs or réseaux
webs, on a space $F$.

######  Definition 7.

  1. (1)

([29, p.715]) A web $\mathcal{W}$ in $F$ is of type (c) if, given any strand
$(W_{k})$ of $\mathcal{W}$ and any sequence $(x_{k})$ with $x_{k}\in W_{k}$
for each $k\in\mathbb{N}$, the series $\sum_{k=1}^{\infty}x_{k}$ converges in
$F$. If in addition the convergent series satisfy
$\sum_{r=k+1}^{\infty}x_{r}\in W_{k-1}$ for each $k\geq 2$, then $\mathcal{W}$
is tight.

  2. (2)

([13, p. 14 - 15], [12, p. 48 - 49]) A réseaux web $\mathcal{R}$ in $F$ is of
type $\mathcal{C}$ if given any sequence $(n_{k})\in\mathbb{N}^{\mathbb{N}}$,
and any $x_{k}\in A_{n_{1}n_{2}\cdots n_{k}}$ for each $k\in\mathbb{N}$, there
exists a sequence $(\lambda_{k})$ of positive numbers, such that the series
$\sum_{k=1}^{\infty}\mu_{k}x_{k}$ converges in $F$ for all $\mu_{k}$ for which
$0\leq|\mu_{k}|\leq\lambda_{k}$, $k\in\mathbb{N}$. If in addition the
convergent series satisfy $\sum_{k=p}^{\infty}\mu_{k}x_{k}\in
A_{n_{1}n_{2}\cdots n_{p}}$, for any $p\in\mathbb{N}$, then $\mathcal{R}$ is
strict.

It is assumed here that all sets in a web or réseaux web are absolutely
convex. As observed in [29, p. 725], a web of type (c) is equivalent to a
réseaux of type $\mathcal{C}$, and a tight web is equivalent to a strict web.
A space with a completing web or réseaux web is typically referred to as a
webbed space, and spaces with tight webs or strict réseaux webs are referred
to as strictly webbed spaces or having a strict web. This is how we will
generally refer to webs in the sequel. We need a few more properties of webs:

######  Definition 8.

A web $\mathcal{W}$ on a space $E$ is:

  1. (1)

Compatible [30, p. 156] if, given any zero neighborhood $V$ in $E$, and any
strand $(W_{k})$ of $\mathcal{W}$, there exists $K\in\mathbb{N}$ such that
$W_{K}\subset V$. Hence, in this case, $(\forall k\geq K)\,W_{k}\subset V$.

  2. (2)

Sequential [23, Def. 6, p. 475] if it is compatible, and given any null
sequence $(x_{n})$ in $E$, there is a strand $(W_{k})$ of $\mathcal{W}$ for
which, given any $k\in\mathbb{N}$, there is $N_{k}\in\mathbb{N}$ such that
$x_{n}\in W_{k}$, for all $n\geq N_{k}$. In this construction, one assumes the
property of strands $(W_{k})$ of W. Robertson’s webs c.f. [30, p. 156] as
expressed in (*) of Definition 5.

If $E$ is any metrizable space, and $\\{V_{k}:k\in\mathbb{N}\\}$ is a system
of absolutely convex zero neighborhoods such that $(\forall
k\in\mathbb{N})\;2V_{k+1}\subset V_{k}$, then
$\mathcal{W}=\\{V_{k}:k\in\mathbb{N}\\}$ is a sequential web. In particular,
the $k$-th layer of $\mathcal{W}$ consists of the set $V_{k}$. General
references for webs on topological vector spaces include [30], [12], and [29].

###  2.3. Ordered webs and quasi-(LB) spaces.

######  Definition 9.

[36, p. 150] A web $\\{W_{n_{1}n_{2}\cdots
n_{k}}:k,n_{1},n_{2},\cdots,n_{k}\in\mathbb{N}\\}$ is ordered if, given
arbitrary natural numbers $k,r_{1},r_{2},\cdots,r_{k}$, and
$s_{1},s_{2},\cdots,s_{k}$, if $r_{j}\leq s_{j}$ for all $j\in\underline{k}$,
then

$W_{r_{1}r_{2}\cdots r_{k}}\subset W_{s_{1}s_{2}\cdots s_{k}}.$

######  Definition 10.

[36, p. 151] Given $\alpha=(a_{n})\in\mathbb{N}^{\mathbb{N}}$ and
$\beta=(b_{n})\in\mathbb{N}^{\mathbb{N}}$, one puts $\alpha\leq\beta$ if
$a_{n}\leq b_{n}$ for every $n\in\mathbb{N}$. A quasi-(LB) representation on a
space $F$ is a family of Banach disks
$\\{A_{\alpha}:\alpha\in\mathbb{N}^{\mathbb{N}}\\}$ such that:

  1. (1)

$\bigcup\\{A_{\alpha}:\alpha\in\mathbb{N}^{\mathbb{N}}\\}=F$;

  2. (2)

For any $\alpha,\beta\in\mathbb{N}^{\mathbb{N}}$, if $\alpha\leq\beta$, then
$A_{\alpha}\subset A_{\beta}$.

For short, we call such an $F$ a quasi- (LB) space. For a given
$\alpha\in\mathbb{N}^{\mathbb{N}}$, we denote by $F_{A_{\alpha}}$ the
corresponding Banach space $(F_{A_{\alpha}},||\cdot||_{A_{\alpha}})$.

The connection between quasi- $(LB)$ spaces and webs that we will make use of
is given by the following.

######  Theorem 2.

[36, 4.1, p. 153] A locally convex space is a quasi-(LB) space if, and only
if, it has an absolutely convex, ordered, strict web. In that case, the
strands are described by: Given a sequence
$(m_{n})\in\mathbb{N}^{\mathbb{N}}$,

(1) $U_{k}\equiv U_{m_{1}m_{2}\cdots
m_{k}}=\bigcup\\{A_{\alpha}:\alpha=(a_{n})\in\mathbb{N}^{\mathbb{N}},a_{n}=m_{n},n\in\underline{k}\\}.$

Observe that $U_{m_{1}}\supset U_{m_{1}m_{2}}\supset\cdots$, for each
$(m_{n})\in\mathbb{N}^{\mathbb{N}}$.
The proofs in [36] use ordered (strict) réseaux web structures as given in
Definitions 6 and 9. The parts of proofs here that correspond to sequential
webs are based on the (tight) web structures of Definition 5, where sequential
convergence is easy to manage. The following simple example illustrates the
difference between an ordered, tight web and and ordered strict réseaux web,
as being equivalent, but having distinct representations on a space.

######  Example 1.

Tight ordered webs and strict ordered réseaux webs on any nontrivial $(LB)$ \-
space.

Details. Let $E=indlim_{n}(E_{n})$, where, for each $n\in\mathbb{N}$, $B_{n}$
denotes the closed unit ball of the Banach space $E_{n}$. Multiplying by
appropriate scalars if necessary, we can assume that for each
$n\in\mathbb{N}$, one has $B_{n}\subset B_{n+1}$. An ordered, tight web on $E$
is described by setting the $k$th layer to be
$\\{2^{-k}B_{n}:n\in\mathbb{N}\\}$, for each $k\in\mathbb{N}$. On the other
hand, an ordered, strict réseaux web is described as follows: Certainly,
$E=\bigcup_{n=1}^{\infty}nB_{n}$, for the first layer. For the second layer,
one writes each $nB_{n}$ as an increasing countable union of Banach disks.
Subsequent layers are obtained by repeating this process. Using values of the
form $2^{j}$ for appropriate values of $j\in\mathbb{Z}$, one can start with an
ordered tight web on $E$ and construct a strict réseaux web representation,
and visa versa.

######  Definition 11.

[36, p. 160] A space is strictly barrelled if for every ordered, absolutely
convex web on the space, there exists at least one strand for which the
closure of each member is a zero neighborhood.

It is easy to see that every strictly barrelled space is barrelled. A
barrelled space that is not strictly barrelled will be given in section 4.

The following definition is intended to express the general assumption that
linear images or linear inverse images of strands of a certain ordered strict
web structure satisfy a condition of closures being zero neighborhoods. See
[36, p. 155, 157].

######  Definition 12.

Consider spaces $E$ and $F$, and a linear map $T:E\rightarrow F$. Assume
$\\{A_{\alpha}:\alpha\in\mathbb{N}^{\mathbb{N}}\\}$ is a quasi - (LB)
representation in $F$. Given $k,m_{1},m_{2},\cdots,m_{k}\in\mathbb{N}$, put

(2) $U_{m_{1}m_{2}\cdots
m_{k}}=T^{-1}\left(\bigcup\\{A_{\alpha}:\alpha=(a_{n})\in\mathbb{N}^{\mathbb{N}},a_{n}=m_{n},n\in\underline{k}\\}\right).$

The triple $(E,F,T)$ satisfies a strand - neighborhood condition if there
exist a sequence $(r_{k})\in\mathbb{N}^{\mathbb{N}}$ such that for
$U_{k}\equiv U_{r_{1}r_{2}\cdots r_{k}}$,

$\overline{U_{k}}=\overline{U_{r_{1}r_{2}\cdots r_{k}}}\in\mathcal{N}_{0,E},$

for each $k\in\mathbb{N}$.

For the case of of a linear $T:F\rightarrow E$ such that $T$ maps a subspace
$H$ of $F$ onto $E$, one defines [36, p. 157]: Given
$n,m_{1},m_{2},\cdots,m_{n}\in\mathbb{N}$, put

(3) $V_{m_{1}m_{2}\cdots m_{n}}=\bigcup\\{T(A_{\alpha}\cap
H):\alpha=(a_{n})\in\mathbb{N}^{\mathbb{N}},a_{n}=m_{n},n\in\underline{k}\\}.$

We assume there exists $(r_{k})\in\mathbb{N}^{\mathbb{N}}$ such that for
$V_{k}\equiv V_{r_{1}r_{2}\cdots r_{k}}$,

$\overline{V_{k}}=\overline{V_{r_{1}r_{2}\cdots r_{k}}}\in\mathcal{N}_{0,E},$

for each $k\in\mathbb{N}$.

We state Valdivia’s closed graph theorem below, as its use will be important
in the proof of the main result. We use the notation of Definition 12.

Valdivia’s Closed Graph Theorem ([36, 5.11, p. 157, 6.1.5, p. 163]). A linear
map from a Hausdorff locally convex space to a quasi -(LB) space having a
closed graph is continuous if the strand neighborhood condition is satisfied.
In particular, we have, in $E$,

  1. (1)

$\emptyset\neq\overline{U_{k}}^{\circ\,\tau_{E}}=\overline{U}^{\circ\,\tau_{E}}_{r_{1}r_{2}\cdots
r_{k}}\subset U_{r_{1}r_{2}\cdots r_{k}},\;k\in\mathbb{N},$

  2. (2)

$T$ is continuous.

One more result is needed in the proof of Theorem 1:

######  Lemma 1.

[27, Lemma 3, p. 60] Let $T:F\rightarrow E$ be linear, for topological vector
spaces $E$ and $F$. Consider

$F\stackrel{{\scriptstyle\varphi}}{{\rightarrow}}F/T^{-1}(\\{0\\})\stackrel{{\scriptstyle\tilde{T}}}{{\rightarrow}}E,$

where $\varphi$ is the canonical linear quotient map, and $\tilde{T}$ is the
linear map such that $T=\tilde{T}\circ\varphi$. If $G_{T}$ denotes the graph
of $T$, then $G_{T}$ is closed if and only if, $G_{\tilde{T}}$ is closed.

##  3\. Proof of the main result.

Proof of Theorem 1. Our methods here are inspired by those in [8], [9], [16],
and [36]. For the sake of completeness, full details are given here. It
suffices to prove (a) and (b), because the proof of part (c) only requires the
observation that any metrizable space has a sequential web and is otherwise
exactly the same as in [36, 5.6, p. 158].

(a). The linear map $T$ is not assumed to be continuous, and the closed graph
theorem does not apply to $T:F\rightarrow E$ (the assumptions are reversed).
Nevertheless, the arrival at the conclusion can be obtained by taking a detour
through the quotient space $H/T^{-1}(\\{0\\})$ in such a way as to apply
Valdivia’s closed graph theorem. This is done in Step 2. After that, we apply
the assumption of a sequential web to prove the existence of the desired
sequence. By way of the linearity of topologies and functions, it suffices to
consider null sequences in our work.

Step 1. Setting up spaces and linear maps. Let $T$ be a linear map from a fast
closed linear subspace $H$ of $F$ onto $E$, for which the graph $G_{T}$ of $T$
is closed in $H\times E$. Let
$\\{A_{\alpha}:\alpha\in\mathbb{N}^{\mathbb{N}}\\}$ be a quasi-(LB)
representation on $F$. Suppose $E$ has a sequential web denoted by
$\mathcal{Z}$, for which we denote strands by
$(Z_{k})_{k\in\mathbb{N}}=(Z_{k})$. Consider any null sequence $(y_{n})$ in
$E$. We wish to show there exists $\beta\in\mathbb{N}^{\mathbb{N}}$ and a
sequence $(u_{n})$ in $F_{A_{\beta}}\cap H$ such that $(u_{n})$ is fast
convergent to zero in $F$, and $T(u_{n})=y_{n}$, for all $n\in\mathbb{N}$.
Using previous notation, we have

(4) $V_{k}\equiv V_{m_{1}\cdots m_{k}}=\bigcup\\{T(A_{\alpha}\cap
H):\alpha\in\mathbb{N}^{\mathbb{N}},\alpha=(a_{n}),\,a_{n}=m_{n},n\in\underline{k}\\}.$

By the strand-neighborhood assumption, there exists a sequence
$(r_{k})\in\mathbb{N}^{\mathbb{N}}$ such that for all $k\in\mathbb{N}$,
$\overline{V}_{r_{1}\cdots r_{k}}\in\mathcal{N}_{0,E}$.

We proceed with a factoring through the null space $T^{-1}(\\{0\\})$, as
follows: Denote by $\varphi$ the canonical quotient map from $H$ onto
$H/T^{-1}(\\{0\\})$. Next, let $S:E{\longrightarrow}H/T^{-1}(\\{0\\})$ be the
bijective linear map such that $T=S^{-1}\circ\varphi$. Observe that for all
$\alpha\in\mathbb{N}^{\mathbb{N}},\,\varphi(A_{\alpha}\cap
H)=S(T(A_{\alpha}\cap H))$. Because $H$ is a fast closed linear subspace of
$F$, for every $\alpha\in\mathbb{N}^{\mathbb{N}}$, $\overline{H\cap
F_{a_{\alpha}}}^{F}=H\cap F_{a_{\alpha}}$, hence, $\\{A_{\alpha}\cap
H:\alpha\in\mathbb{N}^{\mathbb{N}}\\}$ is a quasi- (LB) representation in $H$.
By the continuity of $\varphi$ from $H$ onto $H/T^{-1}(\\{0\\})$, an
application of [36, 3.1, p. 151] guarantees that $\\{\varphi(A_{\alpha}\cap
H):\alpha\in\mathbb{N}^{\mathbb{N}}\\}$ is a quasi- (LB) representation in
$H/T^{-1}(\\{0\\})$.

${H\leq
F}$${E}$${H/T^{-1}(\\{0\\})}$$\scriptstyle{T=\,S^{-1}\circ\;\varphi}$$\scriptstyle{\varphi}$$\scriptstyle{S}$$\scriptstyle{S^{-1}}$

Step 2. Applying Valdivia’s closed graph theorem to the linear map
$S:E\rightarrow H/T^{-1}(\\{0\\})$. We apply Lemma 1 to $T$, using $H$ in
place of $F$, and $\tilde{T}=S^{-1}$: The assumption that the graph of $T$ is
closed in $H\times E$ implies that the graph of $S^{-1}$ is closed in
$H/T^{-1}(\\{0\\})\times E$. Because $S$ is bijective, we conclude that the
graph of $S$ is closed as well. Consider the linear map $S:E\rightarrow
H/T^{-1}(\\{0\\})$ with $H/T^{-1}(\\{0\\})$ being a quasi- (LB) space with
representation $\\{\varphi(A_{\alpha}\cap
H):\alpha\in\mathbb{N}^{\mathbb{N}}\\}$. If we verify the strand -
neighborhood condition $S$, then Valdivia’s closed graph theorem can be
applied to $S$. Given

$\displaystyle C_{m_{1}\cdots m_{k}}$
$\displaystyle=\bigcup\\{\varphi(A_{\alpha}\cap
H):\alpha\in\mathbb{N}^{\mathbb{N}},\alpha=(a_{n}),\,a_{n}=m_{n},n\in\underline{k}\\}$
$\displaystyle=\bigcup\\{S(T(A_{\alpha}\cap
H)):\alpha\in\mathbb{N}^{\mathbb{N}},\alpha=(a_{n}),\,a_{n}=m_{n},n\in\underline{k}\\},$

put

$U_{m_{1}\cdots m_{k}}=S^{-1}(C_{m_{1}\cdots m_{k}}).$

By (3) of Definition 12, we obtain

$\displaystyle S^{-1}(C_{m_{1}\cdots
m_{k}})=S^{-1}\left(\bigcup\\{\varphi(A_{\alpha}\cap
H):\alpha\in\mathbb{N}^{\mathbb{N}},\alpha=(a_{n}),\,a_{n}=m_{n},n\in\underline{k}\\}\right)$
$\displaystyle=\bigcup\\{T(\varphi(A_{\alpha}\cap
H):\alpha\in\mathbb{N}^{\mathbb{N}},\alpha=(a_{n}),\,a_{n}=m_{n},n\in\underline{k}\\}=V_{k}.$

Therefore, with the same $(r_{k})\in\mathbb{N}^{\mathbb{N}}$ as in the
hypotheses,

$\overline{V}^{\,\tau_{E}}_{k}=\overline{V}^{\,\tau_{E}}_{r_{1}r_{2}\cdots
r_{k}}\in\mathcal{N}_{0,E},$

for all $k\in\mathbb{N}$. It now follows that the strand-neighborhood
condition is satisfied for $S$, and we invoke Valdivia’s closed graph theorem
to conclude:

  1. (i).

$\emptyset\neq\overline{V_{k}}^{\circ\,\tau_{E}}=\overline{V}^{\circ\,\tau_{E}}_{r_{1}r_{2}\cdots
r_{k}}\subset V_{r_{1}r_{2}\cdots r_{k}},\;k\in\mathbb{N}$,

  2. (ii).

$S$ is continuous.

In particular, each $V_{r_{1}r_{2}\cdots r_{k}}$ is a zero neighborhood in
$E$.

Step 3. It is now time to apply the assumption of the sequential web for the
null sequence $(y_{n})$ in $E$. Let $(Z_{k})$ of $\mathcal{Z}$ be as in the
hypotheses. Given $k\in\mathbb{N}$, find $j_{k}>k$ such that
$2^{-j_{k}}<k^{-1}$, and observe that

$Z_{j_{k}+k}\subset 2^{-1}Z_{j_{k}+k-1}\subset
2^{-2}Z_{j_{k}+k-2}\subset\cdots\subset 2^{-k}Z_{k}\subset k^{-1}Z_{k},$

as a consequence of the property $Z_{k+1}\subset\frac{1}{2}Z_{k}$. Next, the
sequential property of $\mathcal{Z}$ implies the existence of a positive
integer $N_{j_{k}}$ such that for all $n\geq N_{j_{k}}$, $y_{n}\in
Z_{j_{k}+k}\subset k^{-1}Z_{k}$.

Define, for all $k\in\mathbb{N}$, $d_{n}=k,N_{j_{k}}\leq n<N_{j_{k+1}}$.
Clearly, $\lim_{n\rightarrow\infty}d_{n}=\infty$. We apply the compatibility
of $\mathcal{Z}$ to the zero neighborhoods $V_{l}=V_{r_{1}r_{2}\cdots
r_{l}},\,l\in\mathbb{N}$, as follows. For a fixed $l\in\mathbb{N}$, there
exists $K_{l}\in\mathbb{N}$ such that for all $k\geq K_{l}$, $Z_{k}\subset
V_{l}$. Next, by the construction at the beginning of this step, there exists
$N_{j_{K_{l}}}\in\mathbb{N}$ such that for all $n\geq N_{j_{K_{l}}}$,
$d_{n}y_{n}\in V_{l}$. The surjectivity of $T$ allows us to find a sequence
$(w_{n})$ in $H$, such that for all $n\geq N_{1}\;w_{n}\in H$ such that
$T(w_{n})=d_{n}y_{n},\,N_{j_{k}}\leq n<N_{j_{k+1}}$, for each
$k\in\mathbb{N}$. By [9, 5.21 (i) p.43] with $p=1$, there exists
$\beta\in\mathbb{N}^{\mathbb{N}}$ such that for each $n\in\mathbb{N}$,
$w_{n}\in F_{A_{\beta}}$, where $F_{A_{\beta}}$ is the Banach space
corresponding to the Banach disk $A_{\beta}$, having
$\\{m^{-1}:m\in\mathbb{N}\\}$ as a base for a system of zero neighborhoods.
Put $u_{n}=\frac{1}{d_{n}}w_{n}$. Then for all $n\geq N_{j_{K_{1}}},w_{n}\in
A_{\beta}$, hence, $u_{n}\in\frac{1}{d_{n}}A_{\beta}$, and because
$\lim_{n\rightarrow\infty}d_{n}=\infty$, $(u_{n})$ is fast convergent to zero
in $F_{A_{\beta}}$. Finally, $u_{n}\in H\cap F_{A_{\beta}}$, and

$T(u_{n})=S^{-1}\left(\varphi(\frac{1}{d_{n}}w_{n})\right)=\frac{1}{d_{n}}d_{n}y_{n}=y_{n},$

finishing the proof of part (a).

(b). Let $E$ be any space such that each precompact set is contained in the
absolutely convex hull of a null sequence. The goal is to prove that for any
precompact $B\subset E$ , there exists $\beta\in\mathbb{N}^{\mathbb{N}}$ and a
set $M\subset F_{A_{\beta}}\cap H$ such that $M$ is precompact in
$F_{A_{\beta}}$, and $T(M)=B$. By the assumption on precompact sets, there is
a null sequence $(y_{n})$ in $E$ such that
$\overline{absconv\\{y_{n}\\}}^{\tau_{E}}\supset 2B$. Let
$\beta\in\mathbb{N}^{\mathbb{N}}$ and $(u_{n})$ in $H\cap F_{A_{\beta}}$
corresponding to $(y_{n})$, be as in the proof of part (a). Define
$D=\overline{absconv\\{\frac{1}{2}u_{n}:n\in\mathbb{N}\\}}^{||\cdot||_{A_{\beta}}}$.
Because the subspace $H$ is fast closed in $F$, and the injection
$(F_{A_{\beta}},||\cdot||_{A_{\beta}})\hookrightarrow F$ is continuous, we
obtain:

$D\subset\overline{H\cap
F_{A_{\beta}}}^{||\cdot||_{A_{\beta}}}\subset\overline{H\cap
F_{A_{\beta}}}^{F}=H\cap F_{A_{\beta}}\subset H.$

In particular, intersecting with $H$, the following is continuous:

(**) $H\cap F_{A_{\beta}}\hookrightarrow
H\stackrel{{\scriptstyle\varphi}}{{\rightarrow}}H/T^{-1}(\\{0\\}).$

That $\left(F_{A_{\beta}},||\cdot||_{A_{\beta}}\right)$ is a Banach space
implies $D\subset F_{A_{\beta}}$ is compact. The continuity of the linear
mappings in (**) ensures that $\varphi(D)$ is compact in $H/T^{-1}(\\{0\\})$.
Next, the graph of $T$ being closed in $H\times E$ tells us that
$H/T^{-1}(\\{0\\})$ is Hausdorff ([30, prop. V.1, p.77]). Thus, $\varphi(D)$
is closed in $H/T^{-1}(\\{0\\})$. We use the continuity of $S$ from part (a)
to observe that the set $S^{-1}(\varphi(D))$ is closed in $E$. Meanwhile,
$S^{-1}(\varphi(D))=T(D)\supset B$, which means $T^{-1}(B)\subset D$. In order
to ensure we capture exactly $B$, we intersect; that is, we let
$M=T^{-1}(B)\cap D$. This finishes the proof. $\Box$

The assumption on $E$ in part (b) includes metrizable spaces by way of a
variant of the Banach - Dieudonné Theorem ([25, 10.(3)p. 273]). As it also
follows, this collection of spaces includes any sequentially retractive
inductive limit of metrizable spaces, which of course, need not be metrizable:
Witness any strict ([26, 0.3.1, p. 2]) proper $(LB)$ space [26, 8.5.18, p.
288], or the space $\mathcal{D}$ of test functions from distribution theory.
Moreover, we will see (Example 3 in the next section) that such spaces are not
strictly barrelled. At the same time, part (c) does not assume the space $E$
is strictly barrelled. The following corollary indicates that Theorem 1 here
properly extends Valdivia’s lifting theorem to more general spaces, including
those often seen in applications.

######  Corollary 1.

Let $E=\lim_{n}(E_{n})$ be a sequentially retractive inductive limit of
metrizable spaces, with quasi-(LB) space $F$, $H$, and $T$ as in Theorem 1. If
the images of the quasi- (LB) representation of $H$ under $T$ satisfy the
strand - neighborhood condition, then every convergent sequence in $E$ is the
image under $T$ of a sequence in $H$ that is fast convergent in $F$, and each
precompact subset of $E$ is the image under $T$ of a subset of $H$ that is
fast precompact in $F$.

Proof: It suffices to prove that such a space $E$ has a sequential web. That
proof is found in [23, Prop. 9, p. 476]. $\Box$.

For the corollary below, we consider the space $\mathcal{D}_{\Gamma}^{\prime}$
of distributions having their wavefront sets in a specified closed cone
$\Gamma$, equipped with its normal topology, as defined in [11, p. 1350].
Apart from its importance in distribution theory, the space
$\mathcal{D}_{\Gamma}^{\prime}$ is interesting in its own right, due to the
combination of topological vector space properties it does, and does not,
possess. A lucid description of several such properties can be found in [11].
A few such properties worth mentioning here are that
$\mathcal{D}_{\Gamma}^{\prime}$ is: a normal space of distributions, complete,
nuclear, but is neither metrizable nor bornological (its strong dual,
$\mathcal{E}_{\Lambda}^{\prime}$, is an incomplete inductive limit). Moreover,
$\mathcal{D}_{\Gamma}^{\prime}$ is not barrelled [11, Cor. 36, p. 1376].
Nonetheless, it turns out that $\mathcal{D}_{\Gamma}^{\prime}$ satisfies the
properties of the range space of Theorem 1:

######  Corollary 2.

Suppose $F$ is a quasi-(LB) space and $T$ is a linear map from a fast closed
subspace $H$ of $F$ onto $\mathcal{D}_{\Gamma}^{\prime}$, for which the graph
of $T$ is closed in $H\times\mathcal{D}_{\Gamma}^{\prime}$. If the images of
the quasi- (LB) representation of $H$ under $T$ satisfy the strand -
neighborhood condition, then for any sequence $(y_{n})$ converging to 0 in
$\mathcal{D}_{\Gamma}^{\prime}$, there exists
$\beta\in\mathbb{N}^{\mathbb{N}}$ and a sequence $(u_{n})$ in
$F_{A_{\beta}}\cap H$ such that $(u_{n})$ is fast convergent, and
$T(u_{n})=y_{n}$, for all $n\in\mathbb{N}$, where $A_{\beta}$ is from the
quasi - (LB) space representation on $F$.

Proof: We only need to prove that $\mathcal{D}_{\Gamma}^{\prime}$ has a
sequential web. As a projective limit of a sequence of strictly webbed spaces,
$\mathcal{D}_{\Gamma}^{\prime}$ is strictly webbed by hereditary properties,
namely, that the strong dual of a Fréchet space as well as projective limits
of sequences of such spaces are strictly webbed, [30]. Moreover,
$\mathcal{D}_{\Gamma}^{\prime}$ satisfies the Mackey convergence condition,
[11, Lemma 21, p. 1363], which allows us to apply [23, Thm. 18, p. 481]: a
strictly webbed (locally) complete space satisfying the Mackey convergence
condition has a sequential web. $\Box$.

##  4\. Examples

In [36, Sect. 6, p. 160], the domain space of the closed graph theorem is
assumed to be strictly barrelled. Likewise, in the lifting theorem [36, 5.12,
p. 157, 6.4, p.162] the range space is assumed to be metrizable and strictly
barrelled. The following examples reveal that the collection of domain spaces
in Valdivia’s closed graph theorem (respectively, range spaces of Theorem 1),
are in fact strictly larger than the collection of strictly barrelled
(respectively, metrizable strictly barrelled) spaces.

######  Example 2.

Some spaces having a sequential web that are not strictly barrelled:

  1. (a)

Any metrizable space that is not strictly barrelled.

  2. (b)

The space $(l_{1},\sigma(l_{1},l_{\infty}))$.

Details: (a): Obvious.

(b): Thanks to Shur’s theorem (c.f. [14, p.85]), weakly convergent sequences
are norm convergent in $l_{1}$. Thus, a sequential web is given by
$\mathcal{Z}=\\{2^{-n}B:n\in\mathbb{N}\\}$, where $B$ is the closed unit ball
of $l_{1}$. The space $(l_{1},\sigma(l_{1},l_{\infty}))$ is, of course,
neither metrizable nor barrelled. Incidentally, for any $1<p,q<\infty$ as
conjugate exponents, $(l_{p},\sigma(l_{p},l_{q}))$ is neither (strictly)
barrelled, nor does it have a sequential web.

The next two examples show that the properties of having a sequential web and
being strictly barrelled are distinct. In addition, there are barrelled spaces
that have sequential webs but are not strictly barrelled.

######  Example 3.

Barrelled spaces having a sequential web that are not strictly barrelled.

Details: Let $E$ be any proper sequentially retractive inductive limit of
metrizable barrelled spaces $E_{n}$. As an inductive limit of barrelled
spaces, $E$ is also barrelled. Consider the web formed by the strands
consisting of sets of the form $\\{2^{-k}V_{k,n}:k\in\mathbb{N}\\}$, where,
for each fixed $n\in\mathbb{N}$, $\\{V_{k,n}:k\in\mathbb{N}\\}$ is a
decreasing sequence of absolutely convex sets forming a system of zero
neighborhoods in $E_{n}$ such that for every $k\in\mathbb{N},V_{k+1,n}\subset
2^{-1}V_{k,n}$. Sequential retractivity tells us that this construction forms
a sequential web on $E$. This web is also absolutely convex, ordered, and has
the property that the closure of every set in every strand has empty interior
in $E$. In particular, any strict $(LB)$\- space is barrelled, is quasi- (LB),
has a sequential web, and yet is not strictly barrelled.

For the next example, note that by [23, Thm 12, p. 477], any space with a
sequential web satisfies the Mackey convergence condition.

######  Example 4.

A strictly barrelled space without a sequential web.

Details: Let $I$ be the index set consisting of all increasing, unbounded
sequences of positive real numbers. Put $E=\Pi_{\alpha\in I}\mathbb{R}$. As an
uncountable product of the Banach space $\mathbb{R}$, $E$ is a Baire space, by
[4, Chap. III, Sect. 7]. Thus, $E$ is totally barrelled, hence strictly
barrelled, by [36, 6.17, p. 160]. On the other hand, in [26, 5.1.32, p. 159],
it is shown that $E$ does not satisfy the Mackey convergence condition,
guaranteeing that $E$ cannot have a sequential web.

We finish this section with two observations of hereditary properties of
spaces having sequential webs, that are relevant to Theorem 1. In particular,
in part (b) of the proposition below strict barrelledness is replaced by the
property of having a sequential web, which as we have just seen, is a distinct
collection of spaces. Compare to the strict barrelledness result of [36, 6.16,
p. 160].

######  Proposition 1.

The property of having a sequential web is preserved under:

  1. (a)

Linear subspaces.

  2. (b)

Finitely many products.

Proof: For (a), let $\mathcal{Z}$ be a sequential web on $E$. If $L$ is a
subspace of $E$, define $\mathcal{Z}_{L}$ to be the set of all $Z\cap L$, as
$Z$ runs through the sets of $\mathcal{Z}$. The property of having a
sequential web follows immediately.
Regarding (b), it suffices to prove the case for the product of two spaces $E$
and $F$, the general finite product case following by induction. Denote by
$\mathcal{Z}_{E}$ and $\mathcal{Z}_{F}$ sequential webs in $E$ and $F$,
respectively. For each $k\in\mathbb{N}$, define the $k$th layer in $E\times F$
by the products of the sets in the $k$th layer in $\mathcal{Z}_{E}$ with the
sets in the $k$th layer of $\mathcal{Z}_{F}$. It is easy to verify that this
web is compatible with the topology of the product $E\times F$. Now suppose
$(w_{n})=(x_{n},y_{n})$ is any null sequence in $E\times F$. By assumption,
there is a strand $(Z_{k})$ in $\mathcal{Z}_{E}$ for which, given any
$k\in\mathbb{N}$, there exists $N_{k}$ in $\mathbb{N}$ such that $x_{n}\in
Z_{k}$ for all $n\geq N_{k}$. Similarly, there is a strand $(S_{k})$ in
$\mathcal{Z}_{F}$ satisfying that, given any $k\in\mathbb{N}$, there exists
$M_{k}$ in $\mathbb{N}$ such that $y_{n}\in S_{k}$ for all $n\geq M_{k}$.
Putting $J_{k}=\max\\{N_{k},M_{k}\\}$, for each $k\in\mathbb{N}$, one has for
all $k\in\mathbb{N}$, and for all $n\geq J_{k}$:

$w_{n}=(x_{n},y_{n})\in Z_{k}\times S_{k}.\;\;\;\;\;\Box$

##  5\. Concluding remarks

Some implications for further research regarding the application of Valdivia’s
closed graph theorem and the lifting theorem proved here are outlined.

Applying Valdivia’s closed graph theorem to spaces that are not barrelled. The
relevant examples presented here, including non-barrelled spaces that need not
have sequential webs, could appear as domain spaces for Valdivia’s closed
graph theorem.

Applying Theorem 1 to normed or metrizable spaces that are not barrelled. It
seems some of the easiest examples of spaces having a sequential web and are
not strictly barrelled turn out to be metrizable and non- barrelled. Far from
being “exotic”, normed and metrizable spaces that are not barrelled are rather
common. Examples of such spaces include:

  1. (1)

On every infinite dimensional Banach space there exists a finer, non-barrelled
norm [26, 4.6.7 (iv), p. 131].

  2. (2)

Metrizable, non-normed non-barrelled spaces can be formed by countable
products of non-barrelled normed spaces, [26, 4.2.5, p. 103].

  3. (3)

From [19, Chapt. 6], we have: Let $l_{0}^{\infty}(\Sigma)$ denote the space of
simple functions on an algebra $\Sigma$ of sets, endowed with a natural norm
arising from the Minkowski functional. Then, every separable infinite-
dimensional subspace of $l_{0}^{\infty}(\Sigma)$ is a normed, non-barrelled
space [19, 6.3.2, p. 127]. Furthermore, various algebras $\Sigma$ are
constructed in [19, p. 131 - 135] for which $l_{0}^{\infty}(\Sigma)$ is not
barrelled.

General non-barrelled spaces. Our results here, as well as Valdivia’s closed
graph theorem appear pertinent to the study of general, not necessarily
metrizable, non-barrelled spaces like: Any infinite dimensional Banach space
with its weak topology, and as we have seen, as well as
$\mathcal{D}_{\Gamma}^{\prime}$. Other such spaces, such as tensor products of
spaces related to $\mathcal{D}_{\Gamma}^{\prime}$, as in [5], could also be
worth studying. More examples of non-barrelled spaces and results related to
such spaces can be found in [6], [18], [32], [33], [34], [37], to mention a
few instances.

Developing results in other contexts. The webbed and strictly webbed space
results initially proved for locally convex spaces over the fields of
$\mathbb{R}$ or $\mathbb{C}$ have since been developed for general topological
vector spaces, for locally convex spaces over non-Archimedean fields, for
bornological vector spaces, and for locally convex topological
$\widetilde{\mathbb{C}}$ \- modules, resulting in useful applications of each
those versions. See [1], [21], [22], and [38], respectively. Valdivia’s closed
graph, open mapping, and lifting theorems have been developed for general
topological vector spaces, as in [8], [9], [16]. Beyond that, no further
developments have been made. Such research regarding Valdivia’s closed graph
theorem and the lifting theorem obtained here, in contexts of locally convex
spaces over non-Archimedean fields, bornological vector spaces, and locally
convex topological $\widetilde{\mathbb{C}}$ \- modules, has potential for
correspondingly useful applications.

######  Acknowledgement 1.

I am grateful to several anonymous referees who provided important comments
that significantly improved the quality of this paper.

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# Controllable energy angular spectrum method

Fan Wang  1,* Tomoyoshi Shimobaba  1 and Takashi Kakue1  Tomoyoshi Ito1
1Graduate School of Engineering, Chiba University, Yayoi-cho 1-33, Inage-ku,
Chiba 263-8522, Japan<EMAIL_ADDRESS>
###### Abstract

A controllable energy method, which considers the undersampling issue of the
transfer function and valid spectral energy of a source signal, is proposed to
implement angular spectrum diffraction calculation in near and far fields. The
proposed method provides an optimized frequency boundary $f_{CE}$ within which
it always keeps controllable energy to be diffracted. The controllable energy
angular spectrum method significantly reduces the number of samples while
having the same accuracy as previous angular spectrum methods, implying a
higher calculation efficiency. The new perspective of analyzing spectral
energy is shown to improve the performance of relevant diffraction
calculations.

††journal: osajournal††articletype: Research Article

##  1 Introduction

Scalar diffraction theory plays a pivotal role in optical information
processing by providing the fundamental computational method of light wave
propagation, called the Rayleigh–Sommerfeld diffraction formula [1]. In
numerical calculations, the Rayleigh–Sommerfeld method is generally classified
into three for implementation [2, 3, 4]: the single Fourier transform, three
Fourier transform based on the impulse response, also known as the
convolution-based method, and double Fourier transform based on a transfer
function, also known as the angular spectrum (AS) method. The single Fourier
transform and three Fourier transform (or convolution-based) methods are based
on the Fresnel approximation, whereas the double Fourier transform (or AS)
method is strictly derived from the Rayleigh–Sommerfeld formula. The single
Fourier transform method produces a target plane with a different size from
the source plane because the sampling interval changes with the diffractive
distance, whereas the convolution-based and AS methods maintain the same size
of two parallel planes. In most cases, we expect that diffraction calculations
yield a target plane of the same size as the source plane. Consequently, the
AS method has become significant in the field of numerical diffraction
computation.

However, the AS method has a well-known limitation that produces considerable
aliasing distortion due to undersampling at long-distance diffraction, which
has been revealed in many studies [4, 5, 6, 7, 8]. Over the past decade,
scholars have proposed their own approaches on how to enable the AS method to
be performed over a wide diffraction range, such as the band-limited AS method
by Matsushima [5], wide-window AS method by Yu [6], wide-range AS method by
Kim [7], band-extended AS method [8] and adaptive sampling AS method [9] by
Zhang. In addition, by drawing on the AS diffraction method at long-distance,
scholars have further developed methods applied in the off-axis destination
plane[10, 11], oblique illumination [12] and the polygon-based holograms of
large size [13]. Therefore, an accurate and effective AS method is required to
utilize diffraction calculations widely.

Figure 1: Schematic of a plane (a) and a 1D aperture (b) diffraction .

##  2 Sampling analysis of previous AS methods

Figure. 1(a) shows a diffraction propagation scheme from the source plane to
the target plane in the (x,y,z) system. Mathematically, this can be described
using the AS method as follows:

$\begin{split}u(x,y)&=\mathcal{F}^{-1}\left\\{A(f_{x},f_{y})\cdot
H(f_{x},f_{y})\right\\}\\\
A(f_{x},f_{y})&=\mathcal{F}[u_{0}(x,y)]\,,\end{split}$ (1)

where $u_{0}(x,y)$ and $u(x,y)$ are the complex amplitude distribution in the
source and destination window sampled by $N_{x}\times N_{y}$, respectively,
the operators $\mathcal{F}$ and $\mathcal{F}^{-1}$ are the fast Fourier
transform (FFT) and inverse FFT, respectively. $A(f_{x},f_{y})$ and
$H(f_{x},f_{y})$ are also known as angular spectrum and transfer function,
respectively. $H(f_{x},f_{y})$ is given by

$H(f_{x},f_{y})=\exp\left(jkz\sqrt{1-(\lambda f_{x})^{2}-(\lambda
f_{y})^{2}}\right)\,,$ (2)

where $\lambda$ is the wavelength, $k=2\pi/\lambda$, $j=\sqrt{-1}$, $z$ is the
propagation distance, and $(f_{x},f_{y})$ is the frequency coordinates. In the
numerical computation, a considerable error, resulting from the circular
convolution in Eq. (1), can be avoided by doubling the number of samples of
the source window and padding the outer area with zeros, as shown the zero-
padding window of Fig. 1(a), which the reason of zero-padding has been widely
discussed in [4, 5, 9].

For simplicity, the following discussion is based on the one-dimensional (1D)
case, as shown in Fig. 1(b). From Eq.(2), the phase of transfer function,
defined as $\phi(f_{x})=kz\sqrt{1-(\lambda f_{x})^{2}}$ in 1D, increases as
$z$ gets farther and $f_{x}$ gets greater. For satisfying the sampling
theorem, $\phi(f_{x})$ is restricted to

$1/2\pi|\partial\phi/\partial f_{x}|_{max}\leq 1/2\Delta_{f}\,.$ (3)

$\Delta_{f}=1/2N\Delta_{x}$ is the frequency sampling pitch, where
$\Delta_{x}$ is the spatial sampling pitch, and $N$ is the number of samples
of the source window, and $2N$ is the number of samples after double zero-
padding. A critical distance $z_{c}=2N\Delta_{x}^{2}/\lambda$ was clearly
given in [4, 8], which implies that Eq.(2) will be undersampled for greater
than $z_{c}$.

Figure 2: An overview of sampling strategy for all AS methods in 2D. The blue
shade area is the spectral bandwidth, and blue dots are sampling dots used for
calculation. (a) Band-limited AS method, hollow dots mean padding zeros
outside the blue area, (b) Wide-window AS method, (c) Wide-range AS method,
(d) Adaptive-sampling AS method, (e) Band-extended AS method, (f) The proposed
controllable-energy AS method.

Now, we briefly review the development of various long-distance AS methods.
The band-limited AS method [5] presented the first strict derivation of Eq.
(3) that the frequency boundary without undersampling, denoted by $f_{BL}$,
depends on

$f_{BL}=N\Delta_{x}/z\lambda\,.$ (4)

The core of the band-limited AS method is to let $H(f_{x})=0$ for everywhere
$f_{x}>f_{BL}$ without changing the number of samples and sampling interval.
To intuitively illustrate, Fig. 2 shows a two-dimensional (2D) frequency
domain sampling profile for a square window ($2Nx=2Ny=2N$). In Fig. 2(a),
$f_{max}=1/2\Delta_{x}$ is the maximum frequency that can be recorded by the
zero-padding plane, which always satisfies $f_{BL}<f_{max}$ when $z>z_{c}$. At
a very long distance, the calculation results of the band-limited AS method
will gradually fail because the number of valid frequency samples,

$N_{BL}=2f_{BL}/\Delta_{f}=4N\Delta_{x}f_{BL}=4(N\Delta_{x})^{2}/z\lambda,$
(5)

becomes very small. The wild-window AS method [6] forces the valid number of
samples to be $2N$, which means frequency sampling interval will change,
denoted by $\Delta_{f_{WW}}$. The authors stated that the width of the
calculation window size $S_{x}\times S_{y}$ increased with $z$, as shown in
the red dotted line region of Fig. 2(b). The advantage of the wild-window AS
method over the band-limited AS method is that the frequency boundary is
extended from $f_{BL}$ to $f_{WW}$. However, the wild window AS method
requires two convolution operations because FFT is unavailable after changing
the frequency sampling interval to be $\Delta_{f_{WW}}$, which is quite time-
consuming [6, 14]. The wild-range AS method [7] and adaptive-sampling AS
method [9] follow the valid frequency region given in Eq.(4), $f_{BL}$.
However, the difference is that the former forces the number of samples to be
$2N$ by a new interval $\Delta_{f_{WR}}$, as shown in Fig. 2(c), whereas the
latter holds the original sampling interval $\Delta_{f}$ constant and uses
only the valid number of samples $N_{BL}$ to calculate, as shown in Fig. 2(d).
Owing to the change in sampling interval (in the wild-range AS method ) and
the change in sampling number (in the adaptive-sampling AS method), both
employ the nonuniform-FFT (NUFFT) technology [15, 16] instead of FFT, and
their computational efficiency is better than that of the wild-window AS
method but their calculational quality is similar to that of the band-limited
AS method. The band-extended AS method [8] significantly improves on all
previous methods and gives a comprehensive analysis of the boundary frequency

$f_{BE}=\sqrt{N/2\lambda z}\,,$ (6)

which is greater than $f_{BL}$, as shown in Fig. 2(e). In fact, the band-
extended AS method provides the same boundary as the wild-window AS method,
i.e., $f_{BE}=f_{WW}$ and $\Delta_{f_{BE}}=\Delta_{f_{WW}}$, although the
wild-window AS method is based on the concept of widening window size.
However, the most significant improvement of the band-extended AS method is
the use of NUFFT under the frequency region of $f_{BE}$, which makes
calculations more economical than in the wild-window AS method and makes
results more accurate than in the band-limited, wild-range, and adaptive-
sampling AS methods. Table 1 summaries the differences between these AS
methods, where the texts highlighted in red means to keep parameters constant,
whereas the texts highlighted in green mean to adjust parameter as the
distance changes. The wild-window and band-extended AS methods have better
quality as distance increases due to their wide band, whereas the adaptive-
sampling AS method has the lowest workload due to the smallest number of
samples.

[b]  Methods1 Boundary Number Interval Load Quality2 Band-limited AS[5]
$f_{BL}$$\cdots f_{max}$3 $2N$ $\Delta_{f}$ Middle Worse Wild-window AS[6]
$f_{WW}$ $2N$ $\Delta_{f_{WW}}$ High Better Wild-range AS[7] $f_{BL}$ $2N$
$\Delta_{f_{WR}}$ Middle Worse Adaptive-sampling AS[9] $f_{BL}$ $N_{BL}$
$\Delta_{f}$ Low Worse Band-extended AS[8] $f_{BE}$ $2N$ $\Delta_{f_{BE}}$
Middle Better Controllable-energy AS 4 $f_{CE}$ $N_{CE}$ $\Delta_{f_{CE}}$ Low
Better

Table 1: The list of comparisons between all AS methods.

  * 1

The first two methods are implemented by FFT and two convolutions, whereas the
last four are implemented by NUFFT.

  * 2

Change in quality with the longer distance.

  * 3

Boundary for computing is $f_{max}$ but the valid one is $f_{BL}$, zero-
padding between $f_{BL}$ and $f_{max}$.

  * 4

The proposed method that keeps spectral energy proportion constant.

##  3 The proposed method

As discussed above, the abovementioned AS methods emphasized the limitation of
$H(f_{x},f_{y})$ in Eq.(2) in the undersampling problem of long-distance and
derived the boundary frequency based on Eq. (3). However, to date, the effect
of $A(f_{x},f_{y})$ in Eq. (1) has not been closely examined. We argue that,
in addition to the transfer function, spectral energy plays a paramount role
in diffraction. Thus, in this study, we propose a synthetic adaptive sampling
method on the premise that renders the effective energy proportion of spectral
$A(f_{x},f_{y})$ unchanged, named controllable-energy AS method, which has a
frequency boundary $f_{CE}$ always between $f_{BL}$ and $f_{BE}$, as shown in
Fig. 3.

For a 1D aperiodic signal $u_{0}(x)$, its spectrum is $A(f_{x})$. The energy
spectral density is define as $S(f_{x})=|A(f_{x})|^{2}$, and Parseval’s
theorem states that the signal energy in the spatial domain can be expressed
by the energy in the frequency domain, as follows:

$\int_{-\infty}^{\infty}|u_{0}(x)|^{2}\mathrm{d}x=E(f_{x})=2\int_{0}^{f_{x}}S(f_{x})\mathrm{d}f_{x},\qquad
0\leq f_{x}\leq f_{max}$ (7)

which represents the distribution of signal energy as a function of frequency
[17]. Notably, the double integral in Eq. (7) is because the spectrum is
normally symmetrically distributed around the zero-frequency.

Figure 3: Distribution of normalized energy spectral density $S(f_{x})$ (left
axis) and spectral energy $E(f_{x})$ (right axis) with respect to the
frequency of a source window, where the number of samples is $N=1024$ and
aperture sample is $758$.

Figure 3 is an exemplary spectrum distribution of a simple aperture with 758
samples (the entire source window has 1024 samples) obtained directly by FFT.
The black and red curves represent the energy spectral density and spectral
energy, respectively, which both indicate the energy is mainly concentrated in
the low-frequency components. Although the source signal contains the band
$\left[0,f_{max}\right]$, a considerable amount of high-frequency energy is
invalid due to the undersampling issue of the transfer function, e.g., the
band $\left[f_{BE},f_{max}\right]$ in the band-extended AS method, $E_{inv}$,
as shown in the gray shaded area in Fig. 3. Even within the optimized
frequency boundary $f_{BE}$, there are still some redundant sampling points
since the high frequencies extremely close to $f_{BE}$ have generally little
energy and contribute little to the diffractive results.

From Eq. (7), we define the spectral energy within $f_{BE}$ as
$E_{BE}=2\int_{0}^{f_{BE}}S(f_{x})\mathrm{d}f_{x}$ and the spectral energy
within the proposed frequency boundary $f_{CE}$ as
$E_{CE}=2\int_{0}^{f_{CE}}S(f_{x})\mathrm{d}f_{x}$; therefore, we introduce a
user-defined factor $\eta$ to determine the proportion of $E_{CE}$ to
$E_{BE}$, which can be mathematically expressed by

$E_{CE}\geq\eta E_{BE},\qquad 0\leq\eta\leq 1\,.$ (8)

This equation is the kernel of the proposed method, which suggests that the
spectral energy the proposed method used is always proportional to that used
by the band-extended AS method. Thus, we mainly seek the frequency boundary
$f_{CE}$ that just satisfies Eq. (8) from $0$ to $f_{max}$.

To quickly find $f_{CE}$, we can search from $f_{BL}$ given in Eq. (4),
because the energy within $f_{BL}$ is always less than $E_{BE}$. Algorithm 1
provides the pseudocode to search the appropriate value of $f_{CE}$, where
$\lceil\cdot\rceil$ rounds numbers to upper integers of their argument.
Theoretically, $\eta$ can be any value between 0 and 1, which depends on the
quality of results a user wants. Empirically, the quality of results using
$f_{CE}$ as boundary are very consistent with those of using $f_{BE}$ when
$\eta>0.9$. However, a major advantage of the proposed controllable-energy AS
method is that the number of samples is well below that of the band-extended
AS method, but their computational accuracies are the same, which implies that
the computational efficiency of the proposed controllable-energy AS method is
able to well above that of the band-extended AS method.

Algorithm 1 Searching for $f_{CE}$ based on the given value of $\eta$.

1:$\eta\leftarrow 0.995$ $\triangleright$ User-defined value (between 0 and 1)

2:$E_{BE}\leftarrow 2\times\sum\limits_{m=0}^{\lceil
f_{BE}/\Delta_{f}\rceil}|A(m\Delta_{f})|^{2}\cdot\Delta_{f}$

3:$j\leftarrow\lceil f_{BL}/\Delta_{f}\rceil$

4:$E_{CE}\leftarrow
2\times\sum\limits_{m=0}^{j}|A(m\Delta_{f})|^{2}\cdot\Delta_{f}$

5:while $E_{CE}\leq\eta E_{BE}$ do

6: $j\leftarrow j+1$

7: $E_{CE}\leftarrow E_{CE}+2|A(j\Delta_{f})|^{2}\cdot\Delta_{f}$

8:end while

9:return $f_{CE}\leftarrow j\Delta_{f}$

We now analyze the number of samples of the proposed controllable-energy AS
method. Under the constraint of the boundary frequency $f_{CE}$ searched by
Eq. (8), we define a new sampling interval of frequency as
$\Delta_{f_{CE}}=2f_{CE}/N_{CE}$, where $N_{CE}$ is the number of samples of
the controllable-energy AS method, as shown in Fig. 2(f). As a result, Eq. (3)
becomes

$\begin{split}\lambda\frac{zf_{CE}}{\sqrt{1-(\lambda
f_{CE})^{2}}}\leq\frac{N_{CE}}{4f_{CE}}\,,\\\
\Rightarrow{N_{CE}\geq\frac{4\lambda zf^{2}_{CE}}{\sqrt{1-(\lambda
f_{CE})^{2}}}}\,.\end{split}$ (9)

Considering that $(\lambda f_{CE})^{2}\ll 1$ holds in most of cases, and
taking the minimum value of $N_{CE}$, then

$N_{CE}\approx\left\lceil 4\lambda zf^{2}_{CE}\right\rceil\,.$ (10)

As mentioned above, there is $N\Delta_{x}/z\lambda=f_{BL}\leq f_{CE}\leq
f_{BE}=\sqrt{N/2\lambda z}$, which indicates that $N_{BL}\leq N_{CE}\leq 2N$
always exists. Therefore, the effective angular spectrum of the source window
with the sampling interval $\Delta_{f_{CE}}$ and the number of samples
$N_{CE}$ can be discretely calculated as follows:

$A_{CE}(f_{x})={\rm NUFFT_{3}}\\{u_{0}(x)\\},\ $ (11)

where $x=i\Delta_{x},i\in[0,N)$ and $f_{x}=m\Delta_{f_{CE}},m\in[0,N_{CE})$.
The operation ${\rm NUFFT_{3}}\\{\cdot\\}$ denotes NUFFT, which has been
effectively used by many scholars to overcome the limitation when either
sampling interval or number of samples of the input signal does not match
those of the spectrum [7, 8, 9, 18, 19, 20]. Herein, we adopt the type-3 NUFFT
proposed by Lee and Greengard [16]. Based on Eq. (11), the diffraction field
by the proposed controllable-energy AS method can be obtained as follows

$u(x)={\rm NUFFT_{3}}^{-1}\\{A_{CE}(f_{x})\cdot H(f_{x})\\}\,,$ (12)

where the superscript -1 denotes inverse NUFFT. The proposed controllable-
energy AS method can be readily expanded to the 2D case.

##  4 Results

As a comparison, we use the signal-to-noise ratio (SNR) as the evaluation
criterion in this manuscript, which is widely used in the previous methods[5,
6, 7, 8, 9]. Considering the amplitude of diffraction field by the analytical
Rayleigh–Sommerfeld diffraction integral as reference data and the amplitude
of diffraction field by the adaptive-sampling AS, band-extended AS, and
proposed controllable-energy AS methods as evaluation data, Fig. 4 shows the
evaluation results with the parameters of wavelength $\lambda=532{\rm nm}$,
pixel pitch $\Delta_{x}=1\upmu\rm{m}$, source window samples $N=1024$, and
aperture samples $758$, propagating from $z_{c}(=3.8{\rm mm})$ to $500z_{c}$.

From Fig. 4(a), under $\eta=0.995$, the results of the proposed controllable-
energy AS method and the band-extended AS method performed well as the
distance increased, and both were highly consistent, whereas the adaptive-
sampling AS method results worsened at longer distances due to the sharp
shrinking of the boundary frequency $f_{BL}$, see Eq. (4). Moreover, Fig. 4(b)
shows that the number of samples of the proposed controllable-energy AS method
significantly reduced, even when holding $99.5\%$ energy of $E_{BE}$, whereas
the band-extended AS method always maintained $2N$ samples. In the subgraph of
Fig. 4(b), with the current parameters, the proposed controllable-energy AS
method had the same number of samples as the adaptive-sampling AS method
within around $25z_{c}$, which indicated they had the same boundary frequency
because the spectral energy of the adaptive-sampling AS method, $E_{AS}$, was
greater than $0.995E_{BE}$. At a longer distance than $25z_{c}$, the adaptive-
sampling AS method lost more energy so that
$\textbf{while}\mathbf{\cdots}\textbf{do}$ loop in Algorithm 1 was launched to
search the proposed frequency boundary $f_{CE}$. To keep a steady proportion
of energy, $N_{CE}$ needs to automatically increase with distance but not more
than $2N$. How fast $N_{CE}$ increases depends on the value of $\eta$ and how
much high-frequency energy the source signal has.

Figure 4: Results comparison for the adaptive-sampling AS, band-extended AS
and proposed controllable-energy AS methods ($\eta$=$0.995$) between $z=z_{c}$
and $z=500z_{c}$. (a) Comparison of the SNR of amplitude values, and (b)
comparison of the number of samples used for calculation.

Overall, the comparison of these methods revealed that the proposed
controllable-energy AS method had a lighter workload than the band-extended AS
method without a loss of computational accuracy. We further implemented the
proposed method for an arbitrary triangle, as shown in Fig. 5(a). The triangle
lies in the plane $z=0$, with the vertexes $(0.05,0.15,0)$, $(0.1,0.05,0)$,
$(0.2,0.1,0)$, where the unit is millimeter (mm). Figure 5(b) shows the
rasterized triangle on the plane with size $L_{x}\times L_{y}$, which is
parallel to the target plane. The number of pixels and sampling interval were
1024$\times$1024 and 1$\upmu\rm{m}$$\times$$1\upmu\rm{m}$, respectively.

Figures 6(a-d) depict diffraction results of the convolution-based (three
Fourier transforms) method, adaptive-sampling AS method, band-extended AS
method and proposed method with $\eta=0.97$ at the position $z=20z_{c}$,
respectively. For comparing the quality of results, we took the diffraction
filed obtained by the convolution-based method as reference. From Fig.6(b),
the adaptive-sampling AS method induces noises because of the loss of high
frequency information. However, the proposed method offers the same high
accuracy as the band-extended AS method, as shown in Figs.6(c) and (d). Under
the environment of CPU with AMD Ryzen 5 3600 @3.59GHz, the detailed results
are listed in Table 2. Owing to the reduction in the number of sampling, the
elapsed times of the adaptive-sampling AS and the proposed method were 0.86 s
and 1.06 s, respectively, whereas that of the band-extended AS method was 4.33
s, i.e., approximately 4 times acceleration by the proposed method. Note that,
the elapsed time is not linearly related to the number of sampling because the
interpolation required in the NUFFT technology takes a constant expense [15,
16].

Table 2: Comparison of results at different distances and $\eta$. z |  $\eta$ |  |  Adaptive-sampling AS |  Extended-band AS |  Controllable-energy AS (Proposed)
---|---|---|---|---|---
$20z_{c}$ |  0.97 |  SNR(dB) |  30.7 |  52.1 |  51.4
|  |  Time(s) |  0.86 |  4.33 |  1.06
|  |  Sampling pixels |  $102\times 102$ |  $2048\times 2048$ |  $448\times 448$
$2z_{c}$ |  0.99 |  SNR(dB) |  37.3 |  37.4 |  37.2
|  |  Time(s) |  1.3 |  4.4 |  1.00
|  |  Sampling pixels |  $683\times 683$ |  $2048\times 2048$ |  $406\times 406$
Figure 5: An arbitrary triangle with vertexes $(0.05,0.15,0)$mm,
$(0.1,0.05,0)$mm, $(0.2,0.1,0)$mm is used for the diffraction calculation. (a)
Schematic diagram of the triangle in the three dimension coordinates system.
(b) The rasterization of the triangle in the two dimensional plane. Figure 6:
Diffraction results of the triangle obtained by different methods at
$z=20z_{c}$ and $z=2z_{c}$. The baseline of the comparison is the convolution-
based method.

As a discussion, it is worth noting that the spectral energy suggested by the
proposed method is determined by the value of $\eta$ and the spectral energy
of the band-extended method, see Eq. (8), but this does not exclude a broader
application. In Fig. 6(d), the factor of $\eta$ can be further reduced
depending on the user’s requirement for image quality because a processed
image is acceptable to human eyes if its peak-SNR is greater than 30 dB [21],
which implies a smaller number of samples. Moreover, the spectral energy
$E_{BE}$ in Eq. (8) can be replaced with the spectral energy within the
frequency boundary $f_{BL}$ at the closer position, because the accuracy of
the adaptive-sampling method and the band-extended method is almost the same
within a distance of about $10z_{c}$. By doing so, Eq. (8) is rewritten as

$E_{CE}\geq\eta E_{BL},\qquad 0\leq\eta\leq 1\,,$ (13)

where $E_{BL}=2\int_{0}^{f_{BL}}S(f_{x})\mathrm{d}f_{x}$. This indicates that
we can further compress the spectral region of $f_{CE}$. Unsurprisingly, if
$f_{CE}<f_{BL}$, the computational effort will be significantly reduced. For
some intensive calculation cases in the frequency domain, such as the
calculation of the polygon-based hologram [13], the proposed method offers
flexibility in choosing the trade-off between efficiency and accuracy. To
simply confirm this application, Figs. 6(e-h) provide the diffraction results
for the triangle in Fig. 5 at $z=2z_{c}$. Figure 6(h) was obtained based on
Eq. (13) and $\eta=0.99$. From Table 2, the number of sampling by the proposed
method is smaller than that of the adaptive-sampling AS method with the
comparable quality.

##  5 Conclusion

In conclusion, we proposed controllable-energy method to perform diffraction
calculations at near to far distances based on the adaptive-sampling AS method
[9] and the band-extended AS method [8]. The proposed method can effectively
reduce the number of samples compared with the band-extended AS method while
maintaining higher accuracy than the adaptive-sampling AS method. We further
confirmed that the proposed method can be flexibly adapted to the user’s
needs, resulting in more economical computations.

This study verified the effectiveness of the proposed controllable-energy AS
method in the diffraction calculation. Moreover, significantly, it developed a
new perspective of considering the spectrum energy in the diffraction
propagation issues rather than just considering the undersampling problem,
which may provide some insights for related studies.

## Funding

This work was partially supported by JSPS KAKENHI Grant Numbers 19H04132 and
19H01097.

## Acknowledgements

We thank the Ministry of Education, Culture, Sports, Science and Technology
(MEXT) of Japan for their support. And thanks to Dr. Wenhui Zhang and Dr.
Haitong Sui for discussions.

## Disclosures

The authors declare no conflicts of interest.

## Data Availability

Data underlying the results presented in this Letter are not publicly
available at this time but may be obtained from the authors upon reasonable
request.

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