by S. Simons

Abstract

In this paper, we unify the theory of SSD spaces, part of the theory of
strongly representable multifunctions, and the theory of the equivalence of
various classes of maximally monotone multifunctions.

0 Introduction

In this paper, we unify three different lines of investigation: the theory of
SSD spaces as expounded in [11] and [13], part of the theory of strongly
representable multifunctions as expounded in [15] and [4], and the equivalence
of various classes of maximally monotone multifunctions, as expounded in [5].

The purely algebraic concepts of SSD space and $q$–positive set are introduced
in Definition 1.2. These were originally defined in [11], and the development
of the theory was continued in [13]. Apart from the fact that we write “${\cal
P}$” instead of “pos”, we use the notation of the latter of these references.
We show in Lemma 1.9 how certain proper convex functions $f$ on an SSD space
lead to a $q$–positive set, ${\cal P}(f)$. In Definition 1.10, we define the
intrinsic conjugate, $f^{@}$, of a proper convex function on an SSD space, and
we end Section 1 by proving in Lemma 1.11 a simple, but useful, property of
intrinsic conjugates.

In Definition 2.1, we introduce the concept of a Banach SSD space, which is an
SSD space with a Banach space structure satisfying the compatibility
conditions (2.1.1) and (2.1.2). A proper convex function on a Banach SSD space
may be a VZ function, which is introduced in Definition 2.5. Our main result
on VZ functions, established in Theorem 2.9(c,d), is that if $f$ is a lower
semicontinuous VZ function then ${\cal P}(f)$ is maximally $q$–positive,
$f^{@}$ is also a VZ function, and ${\cal P}\big{(}f^{@}\big{)}={\cal P}(f)$.
Lemma 2.7(b) is an important stepping–stone to Theorem 2.9. In Definition 2.12
and Lemma 2.13, we introduce and discuss the properties of various convex
functions on a Banach SSD space and its dual, and show in Theorem 2.15(c) that
if $f$ is a lower semicontinuous VZ function on a Banach SSD then there is a
whole family of VZ functions $h$ associated with $f$ such that ${\cal
P}(h)={\cal P}(f)$.

If $E$ is a nonzero Banach space then it is shown in Examples 1.4, 2.3, and
2.4 that $E\times E^{*}$ is a Banach SSD space under various different norms.
We show in Section 3 how the definitions and results of Section 2 specialize
to this case. Theorem 3.1 extends some concepts and results from [1] and [5].
The definition of VZ function involves the norm of $B$ in an essential way.
Looking ahead, we will see in Theorem 5.3 that there is a large class of norms
on $E\times E^{*}$ for which the classes of VZ functions coincide. This
follows from the analysis in Section 4, which we will now discuss.

In Definition 4.1, we introduce the concept of a Banach SSD dual space, which
is the dual of a Banach SSD space which has an SSD structure in its own right,
satisfying the compatibility conditions (4.1.1) and (4.1.2). In this
situation, a proper convex function on the (original) Banach SSD space may be
an MAS function, which is introduced in Definition 4.8. The main result here
is Theorem 4.9(c), in which we prove that, under the $\widetilde{p}$–density
condition (4.2.1), a function is an MAS function if, and only if, it is a VZ
function. The main stepping stone to Theorem 4.9 is Lemma 4.7, which relies on
Rockafellar’s formula for the conjugate of the sum of two convex functions.

The subtlety of the analysis outlined in the previous paragraph is that
definition of MAS function does not use the norm of $B$ explicitly — it only
uses the knowledge of $B^{*}$. In a certain sense, the analysis of Section 2
is isometric, while the analysis of Section 4 is isomorphic, though it would
be a mistake to push this analogy too far because, despite the fact that the
definition of MAS function does not use the norm of $B$, the conditions
(4.1.2) and (4.2.1) referred to above do use the norm very strongly.

In Section 5, we show how the results of Section 4 specialize to the $E\times
E^{*}$ case. In Theorem 5.5, we show how the negative alignment analysis
introduced in [10, Section 8, pp. 274–280] and [13, Section 42, pp. 161–167]
can be used to obtain, and in some cases strengthen, results from [4] and
[15]. In Theorem 5.8, we generalize some equivalencies from [5, Theorem 1.2].
In particular, we give a proof of the very nice result from [5] that a
maximally monotone multifunction is strongly representable if, and only if, it
is of type (NI).

At one point in this paper, we will use the Fenchel–Moreau theorem for a not
necessarily Hausdorff locally convex space. For the convenience of the reader,
we give a proof of this result in the Appendix, Section 6.

The author would like to thank Constantin Zălinescu for making him aware of
the preprints [4] and [15], and Benar Svaiter for making him aware of the
preprint [5]. He would also like to thank Constantin Zălinescu for some very
perceptive comments on an earlier version of this paper.

1 SSD spaces

We first introduce the concepts of an SSD space and $q$–positive set. As
pointed out in the introduction, these were introduced in [11] and [13]. The
first of these references has a detailed discussion of the finite dimensional
case.

Definition 1.1. If $X$ is a nonzero vector space and $f\colon\
X\to\,]{-}\infty,\infty]$, we write $\hbox{\rm dom}\,f$ for the set
$\big{\\{}x\in X\colon\ f(x)\in\hbox{\tenmsb R}\big{\\}}$. $\hbox{\rm dom}\,f$
is the effective domain of $f$. We say that $f$ is proper if $\hbox{\rm
dom}\,f\neq\emptyset$. We write ${\cal PC}(X)$ for the set of all proper
convex functions from $X$ into $\,]{-}\infty,\infty]$. If $X$ is a nonzero
Banach space, we write ${\cal PCLSC}(X)$ for the set

$\\{f\in{\cal PC}(X)\colon\ f\ \hbox{is lower semicontinuous on}\ X\\},$

and ${\cal PCLSC}^{*}(X^{*})$ for the set

$\\{f\in{\cal PC}(X^{*})\colon\ f\ \hbox{is $w(X^{*},X)$--lower semicontinuous
on}\ X^{*}\\}.$

Definition 1.2. We will say that $\big{(}B,\lfloor\cdot,\cdot\rfloor\big{)}$
is a symmetrically self–dual space (SSD space) (if there is no risk of
confusion, we will say simply “$B$ is an SSD space”) if $B$ is a nonzero real
vector space and $\lfloor\cdot,\cdot\rfloor\colon B\times B\to\hbox{\tenmsb
R}$ is a symmetric bilinear form. We define the quadratic form $q$ on $B$ by
$q(b):={\textstyle{1\over 2}}\lfloor b,b\rfloor$. Let $A\subset B$. We say
that $A$ is $q$–positive if $A\neq\emptyset$ and

$b,c\in A\Longrightarrow q(b-c)\geq 0.$

We say that $A$ is maximally $q$–positive if $A$ is $q$–positive and $A$ is
not properly contained in any other $q$–positive set. We make the elementary
observation that if $b\in B$ and $q(b)\geq 0$ then the linear span
$\hbox{\tenmsb R}b$ of $\\{b\\}$ is $q$–positive.

We now give some examples of SSD spaces and their associated $q$–positive
sets.

Example 1.3. Let $B$ be a Hilbert space with inner product
$(b,c)\mapsto\langle b,c\rangle$ and $T\colon B\to B$ be a self–adjoint linear
operator. Then $B$ is an SSD space with $\lfloor b,c\rfloor:=\langle
Tb,c\rangle$, and then $q(b)={\textstyle{1\over 2}}\langle Tb,b\rangle$. Here
are three special cases of this example:

(a) If, for all $b\in B$, $Tb=b$ then $\lfloor b,c\rfloor:=\langle
b,c\rangle$, $q(b)={\textstyle{1\over 2}}\|b\|^{2}$ and every subset of $B$ is
$q$–positive

(b) If, for all $b\in B$, $Tb=-b$ then $\lfloor b,c\rfloor:=-\langle
b,c\rangle$, $q(b)=-{\textstyle{1\over 2}}\|b\|^{2}$ and the $q$–positive sets
are the singletons.

(c) If $B=\hbox{\tenmsb R}^{3}$ and $T(b_{1},b_{2},b_{3})=(b_{2},b_{1},b_{3})$
then

$\big{\lfloor}(b_{1},b_{2},b_{3}),(c_{1},c_{2},c_{3})\big{\rfloor}:=b_{1}c_{2}+b_{2}c_{1}+b_{3}c_{3},$

and $q(b_{1},b_{2},b_{3})=b_{1}b_{2}+{\textstyle{1\over 2}}b_{3}^{2}$. Here,
If $M$ is any nonempty monotone subset of $\hbox{\tenmsb R}\times\hbox{\tenmsb
R}$ (in the obvious sense) then $M\times\hbox{\tenmsb R}$ is a $q$–positive
subset of $B$. The set $\hbox{\tenmsb R}(1,-1,2)$ is a $q$–positive subset of
$B$ which is not contained in a set $M\times\hbox{\tenmsb R}$ for any monotone
subset of $\hbox{\tenmsb R}\times\hbox{\tenmsb R}$. The helix
$\big{\\{}(\cos\theta,\sin\theta,\theta)\colon\theta\in\hbox{\tenmsb
R}\big{\\}}$ is a $q$–positive subset of $B$, but if $0<\lambda<1$ then the
helix
$\big{\\{}(\cos\theta,\sin\theta,\lambda\theta)\colon\theta\in\hbox{\tenmsb
R}\big{\\}}$ is not.

Example 1.4. Let $E$ be a nonzero Banach space and $B:=E\times E^{*}$. For all
$b=(x,x^{*})$ and $c=(y,y^{*})\in B$, we set $\lfloor b,c\rfloor:=\langle
x,y^{*}\rangle+\langle y,x^{*}\rangle$. Then $B$ is an SSD space and

$q(b)={\textstyle{1\over 2}}\big{[}\langle x,x^{*}\rangle+\langle
x,x^{*}\rangle\big{]}=\langle x,x^{*}\rangle.$

Consequently, if $b=(x,x^{*})\ \hbox{and}\ c=(y,y^{*})\in B$ then

$\langle
x-y,x^{*}-y^{*}\rangle=q(x-y,x^{*}-y^{*})=q\big{(}(x,x^{*})-(y,y^{*})\big{)}=q(b-c).$

Thus if $A\subset B$ then $A$ is $q$–positive exactly when $A$ is a nonempty
monotone subset of $B$ in the usual sense, and $A$ is maximally $q$–positive
exactly when $A$ is a maximally monotone subset of $B$ in the usual sense. We
point out that any finite dimensional SSD space of the form described here
must have even dimension. Thus cases of Example 1.3 with finite odd dimension
cannot be of this form.

Example 1.5. $\hbox{\tenmsb R}^{3}$ is not an SSD space with

$\big{\lfloor}(b_{1},b_{2},b_{3}),(c_{1},c_{2},c_{3})\big{\rfloor}:=b_{1}c_{2}+b_{2}c_{3}+b_{3}c_{1}.$

(The bilinear form $\lfloor\cdot,\cdot\rfloor$ is not symmetric.)

Lemma 1.6. Let $B$ be an SSD space, $f\in{\cal PC}(B)$, $f\geq q$ on $B$ and
$b,c\in B$. Then

$-q(b-c)\leq\Big{[}\sqrt{(f-q)(b)}+\sqrt{(f-q)(c)}\Big{]}^{2}.$

Proof. We can and will suppose that $0\leq(f-q)(b)<\infty$ and
$0\leq(f-q)(c)<\infty$. Let $\sqrt{(f-q)(b)}<\beta<\infty$ and
$\sqrt{(f-q)(c)}<\gamma<\infty$, so that $\beta^{2}+q(b)>f(b)$ and
$\gamma^{2}+q(c)>f(c)$. Then

$\eqalign{\beta\gamma+{\gamma q(b)+\beta
q(c)\over\beta+\gamma}&={\gamma\over\beta+\gamma}\big{(}\beta^{2}+q(b)\big{)}+{\beta\over\beta+\gamma}\big{(}\gamma^{2}+q(c)\big{)}\cr&>{\gamma\over\beta+\gamma}f(b)+{\beta\over\beta+\gamma}f(c)\geq
f\bigg{(}{\gamma b+\beta c\over\beta+\gamma}\bigg{)}\cr&\geq q\bigg{(}{\gamma
b+\beta c\over\beta+\gamma}\bigg{)}={\gamma^{2}q(b)+\gamma\beta\lfloor
b,c\rfloor+\beta^{2}q(c)\over(\beta+\gamma)^{2}}.}$

Clearing of fractions, we obtain

$(\beta+\gamma)^{2}\beta\gamma+(\beta+\gamma)\big{(}\gamma q(b)+\beta
q(c)\big{)}>\gamma^{2}q(b)+\gamma\beta\lfloor b,c\rfloor+\beta^{2}q(c),$

from which $(\beta+\gamma)^{2}\beta\gamma>-\beta\gamma q(b)+\beta\gamma\lfloor
b,c\rfloor-\beta\gamma q(c)=-\beta\gamma q(b-c)$. If we now divide by
$\beta\gamma$, we obtain $(\beta+\gamma)^{2}>-q(b-c)$, and the result follows
by letting $\beta\to\sqrt{(f-q)(b)}$ and $\gamma\to\sqrt{(f-q)(c)}$. □

Remark 1.7. It follows from Lemma 1.6 and the Cauchy–Schwarz inequality that

$-q(b-c)\leq 2(f-q)(b)+2(f-q)(c).$

In the situation of Example 1.4, we recover [15, Proposition 1].

Definition 1.8. If $B$ be an SSD space, $f\in{\cal PC}(B)$ and $f\geq q$ on
$B$, we write

${\cal P}(f):=\big{\\{}b\in B\colon\ f(b)=q(b)\big{\\}}.$

The following result is suggested by Burachik–Svaiter, [1, Theorem 3.1, pp.
2381–2382] and Penot, [7, Proposition 4(h)$\Longrightarrow$(a), pp. 860–861].

Lemma 1.9. Let $B$ be an SSD space, $f\in{\cal PC}(B)$, $f\geq q$ on $B$ and
${\cal P}(f)\neq\emptyset$. Then ${\cal P}(f)$ is a $q$–positive subset of
$B$.

Proof. This is immediate from Lemma 1.6. □

We now introduce a concept of conjugate that is intrinsic to an SSD space
without any topological conditions.

Definition 1.10. If $B$ is an SSD space and $f\in{\cal PC}(B)$, we write
$f^{@}$ for the Fenchel conjugate of $f$ with respect to the pairing
$\lfloor\cdot,\cdot\rfloor$, that is to say,

$\hbox{for all}\ c\in B,\qquad f^{@}(c):=\sup\nolimits_{b\in B}\big{[}\lfloor
b,c\rfloor-f(b)\big{]}.$

Our next result represents an improvement of the result proved in [13, Lemma
19.12, p. 82], and uses a disguised differentiability argument. See Remark
1.12 below for another proof of Lemma 1.11, due to Constantin Zălinescu.

Lemma 1.11. Let $B$ be an SSD space, $f\in{\cal PC}(B)$ and $f\geq q$ on $B$.
Then:

$\leqalignno{a\in{\cal P}(f)\ \hbox{and}\ b\in
B&\quad\Longrightarrow\quad\lfloor b,a\rfloor\leq q(a)+f(b).&(a)\cr a\in{\cal
P}(f)&\quad\Longrightarrow\quad f^{@}(a)=q(a).&(b)}$

Proof. Let $a\in{\cal P}(f)$ and $b\in B$. Let $\lambda\in\,]0,1[\,$. For
simplicity in writing, let $\mu:=1-\lambda\in\,]0,1[\,$. Then

$\eqalign{\lambda^{2}q(b)+\lambda\mu\lfloor
b,a\rfloor+\mu^{2}q(a)&=q\big{(}\lambda b+\mu a\big{)}\leq f(\lambda b+\mu
a)\cr&\leq\lambda f(b)+\mu f(a)=\lambda f(b)+\mu q(a).}$

Thus $\lambda^{2}q(b)+\lambda\mu\lfloor b,a\rfloor\leq\lambda f(b)+\lambda\mu
q(a)$. We now obtain (a) by dividing by $\lambda$ and letting $\lambda\to 0$.
Now let $a\in{\cal P}(f)$. From (a), $b\in B\Longrightarrow\lfloor
a,b\rfloor-f(b)\leq q(a)$, and it follows by taking the supremum over $b\in B$
that $f^{@}(a)\leq q(a)$. On the other hand, $f^{@}(a)\geq\lfloor
a,a\rfloor-f(a)=2q(a)-q(a)=q(a)$, completing the proof of (b). □

Remark 1.12. The author is grateful to Constantin Zălinescu for pointing out
to him the following alternative proof of Lemma 1.11(a). From Lemma 1.6, with
$c$ replaced by $a$, $-q(b)+\lfloor b,a\rfloor-q(a)=-q(b-a)\leq(f-q)(b)$. Thus
$\lfloor b,a\rfloor-q(a)\leq f(b)$, as required.

2 Banach SSD spaces

Definition 2.1. We say that $B$ is a Banach SSD space if $B$ is an SSD space
and $\|\cdot\|$ is a norm on $B$ with respect to which $B$ is a Banach space
with norm–dual $B^{*}$,

${\textstyle{1\over 2}}\|\cdot\|^{2}+q\geq 0\ \hbox{on}\ B$ $None$

and there exists $\iota\in L(B,B^{*})$ such that

$\hbox{for all}\ b,c\in B,\quad\big{\langle}b,\iota(c)\big{\rangle}=\lfloor
b,c\rfloor\hbox{,\quad\big{(}from which }\big{|}\lfloor
b,c\rfloor\big{|}\leq\|\iota\|\|b\|\|c\|\hbox{\big{)}.}$ $None$

Then, for all $d,e\in B$,

$|q(d)-q(e)|={\textstyle{1\over 2}}\big{|}\lfloor d,d\rfloor-\lfloor
e,e\rfloor\big{|}={\textstyle{1\over 2}}\big{|}\lfloor
d-e,d+e\rfloor\big{|}\leq{\textstyle{1\over 2}}\|\iota\|\|d-e\|\|d+e\|.$
$None$

We define the continuous convex functions $g$ and $p$ on $B$ by
$g:={\textstyle{1\over 2}}\|\cdot\|^{2}$ and $p:=g+q$, so that $p\geq 0$ on
$B$. Since $p(0)=0$, in fact

$\inf\nolimits_{B}p=0.$ $None$

Also, for all $d,e\in B$, $|g(d)-g(e)|={\textstyle{1\over
2}}\big{|}\|d\|-\|e\|\big{|}\big{(}\|d\|+\|e\|\big{)}\leq{\textstyle{1\over
2}}\|d-e\|\big{(}\|d\|+\|e\|\big{)}$. Combining this with (2.1.3), for all
$d,e\in B$,

$|p(d)-p(e)|\leq{\textstyle{1\over
2}}\big{(}1+\|\iota\|\big{)}\|d-e\|\big{(}\|d\|+\|e\|\big{)}.$ $None$

(2.1.2) implies that, for all $f\in{\cal PC}(B)$ and $c\in B$,
$f^{@}(c)=\sup\nolimits_{b\in
B}\big{[}\big{\langle}b,\iota(c)\big{\rangle}-f(b)\big{]}=f^{*}\big{(}\iota(c)\big{)}$,
that is to say,

$f^{@}=f^{*}\circ\iota\ \hbox{on}\ B.$ $None$

Remark 2.2. Example 1.3 is a Banach SSD space provided that $\|T\|\leq 1$.
This is the case with (a), (b) and (c) of Example 1.3.

Example 2.3. We now continue our discussion of Example 1.4. We suppose that
$B=E\times E^{*}$ and $\big{(}B,\|\cdot\|\big{)}$ is a Banach SSD space such
that $B^{*}=E^{**}\times E^{*}$, under the pairing

$\langle b,c^{*}\rangle:=\langle x,y^{*}\rangle+\langle
x^{*},y^{**}\rangle\quad\big{(}b=(x,x^{*})\in B,\ c^{*}=(y^{**},y^{*})\in
B^{*}\big{)}.$ $None$

We recall that, for all $(x,x^{*})\in B$, $q(x,x^{*})=\langle x,x^{*}\rangle$.
It is clear that, for all $(x,x^{*})\in B$,
$\iota(x,x^{*}):=(\widehat{x},x^{*})$ where $\widehat{x}$ is the canonical
image of $x$ in $E^{**}$. We note that if $\big{(}B,\|\cdot\|\big{)}$ is a
Banach SSD space and $\|\cdot\|^{\prime}$ is a larger norm on $B$ such that
$\big{(}B,\|\cdot\|^{\prime}\big{)}^{*}=E^{**}\times E^{*}$ then
$\big{(}B,\|\cdot\|^{\prime}\big{)}$ is also a Banach SSD space.

Example 2.4. We now discuss some specific examples of the above concepts. Here
it is convenient to introduce a parameter $\tau>0$. ($\tau$ stands for
“torsion”.) Then$E\times E^{*}$ is a Banach SSD space if we use the norm
$\|(x,x^{*})\|_{1,\tau}:=\textstyle{1\over\sqrt{2}}\big{(}\tau\|x\|+\|x^{*}\|/\tau\big{)}$
or $\|(x,x^{*})\|_{2,\tau}:=\sqrt{\tau^{2}\|x\|^{2}+\|x^{*}\|^{2}/\tau^{2}}$
or
$\|(x,x^{*})\|_{\infty,\tau}:=\sqrt{2}\big{(}\tau\|x\|\vee\|x^{*}\|/\tau\big{)}$.
(These are arranged in order of increasing size.) Then the dual norm of
$\big{(}B,\|\cdot\|_{1,\tau}\big{)}$ is given by
$\|(y^{**},y^{*})\|_{\infty,\tau}:=\sqrt{2}\big{(}\tau\|y^{**}\|\vee\|y^{*}\|/\tau\big{)}$,
the dual norm of $\big{(}B,\|\cdot\|_{2,\tau}\big{)}$ is given by
$\|(y^{**},y^{*})\|_{2,\tau}:=\sqrt{\tau^{2}\|y^{**}\|^{2}+\|y^{*}\|^{2}/\tau^{2}}$,
and the dual norm of $\big{(}B,\|\cdot\|_{\infty,\tau}\big{)}$ is given by
$\|(y^{**},y^{*})\|_{1,\tau}:=\textstyle{1\over\sqrt{2}}\big{(}\tau\|y^{**}\|+\|y^{*}\|/\tau\big{)}$.

Definition 2.5. Let $X$ be a vector space and $h,k\colon
X\to\,]{-}\infty,\infty]$. The inf–convolution of $h$ and $k$ is defined by
$(h\mathop{\nabla}k)(x):=\inf\nolimits_{y\in X}\big{[}h(y)+k(x-y)\big{]}$
($x\in X$). It is clear that

$\inf\nolimits_{X}k=0\quad\Longrightarrow\quad\inf\nolimits_{X}\big{[}h\mathop{\nabla}k\big{]}=\inf\nolimits_{X}h.$
$None$

Now let $\big{(}B,\|\cdot\|\big{)}$ be a Banach SSD space and $f\in{\cal
PC}(B)$. We say that $f$ is a VZ function (with respect to $\|\cdot\|\big{)}$
if

$(f-q)\mathop{\nabla}p=0\ \hbox{on}\ B.$ $None$

It follows from (2.1.4) and (2.5.1) that

$\hbox{if}\ f\ \hbox{is a VZ function with respect to}\ \|\cdot\|\
\hbox{then}\ \inf\nolimits_{B}[f-q]=0.$ $None$

“VZ” stands for “Voisei–Zălinescu”, since (2.5.2) is an extension to Banach
SSD spaces of a condition introduced in [15, Proposition 3].

Definition 2.6. Let $A$ be a subset of a Banach SSD space $B$. We say that $A$
is $p$–dense if, for all $c\in B$, $\inf p(c-A)=0$.

We now come to our main results on Banach SSD spaces. Lemma 2.7(b) is
interesting since it tells us that we can determine whether $f$ is a VZ
function by inspecting ${\cal P}(f)$.

Lemma 2.7. Let $B$ be a Banach SSD space and $f\in{\cal PCLSC}(B)$.

(a) Let $f$ be a VZ function. Then ${\cal P}(f)$ is a $q$–positive subset of
$B$ and

$c\in B\quad\Longrightarrow\quad\hbox{\rm dist}(c,{\cal
P}(f))\leq\sqrt{2}\sqrt{(f-q)(c)}.$ $None$

(b) The following three conditions are equivalent:

(i) $f$ is a VZ function.

(ii) $f\geq q$ on $B$ and, for all $c\in B$ there exists a bounded sequence
$\\{a_{n}\\}_{n\geq 1}$ of elements of ${\cal P}(f)$ such that
$\lim\nolimits_{n\to\infty}p(c-a_{n})=0$.

(iii) $f\geq q$ on $B$ and ${\cal P}(f)$ is $p$–dense.

Proof. (a) (2.5.3) implies that $f\geq q$ on $B$, and so ${\cal P}(f)$ is
defined. Since (2.7.1) is trivial if $c\in B\setminus\hbox{\rm dom}\,f$, we
can and will suppose that $c\in\hbox{\rm dom}\,f$. Let
$\varepsilon\in\,]0,1[\,$. We first prove that there exists a Cauchy sequence
$\\{b_{n}\\}_{n\geq 1}$ such that, for all $n\geq 1$,

$(f-q)(b_{n})\leq(f-q)(c)/16^{n}\quad\hbox{and}\quad\|c-b_{n}\|\leq(1+\varepsilon)\sqrt{2}\sqrt{(f-q)(c)}.$
$None$

Since we can take $b_{n}=c$ if $(f-q)(c)=0$, we can and will suppose that

$\alpha:=\sqrt{(f-q)(c)}>0.$ $None$

Let $\lambda:=\varepsilon/(3+\varepsilon)\in\,]0,1/4[\,$ and write $b_{0}:=c$.
Then we can choose inductively $b_{1},b_{2},\dots\in B$ such that, for all
$n\geq 1$, $(f-q)(b_{n})+p(b_{n-1}-b_{n})\leq\lambda^{2n}\alpha^{2}$. It
follows from this and (2.5.3) that,

$\hbox{for all}\ n\geq 1,\qquad p(b_{n-1}-b_{n})\leq\lambda^{2n}\alpha^{2},$
$None$

and, combining with (2.1.4),

$\hbox{for all}\ n\geq
0,\qquad(f-q)(b_{n})\leq\lambda^{2n}\alpha^{2}\leq\alpha^{2}/16^{n}.$ $None$

Substituting the first inequality of (2.7.5) into Lemma 1.6, for all $n\geq
1$,

$-q(b_{n-1}-b_{n})\leq\Big{[}\sqrt{(f-q)(b_{n-1})}+\sqrt{(f-q)(b_{n})}\Big{]}^{2}\leq(1+\lambda)^{2}\lambda^{2n-2}\alpha^{2}.$

Consequently, since $g(b_{n-1}-b_{n})=p(b_{n-1}-b_{n})-q(b_{n-1}-b_{n})$,
(2.7.4) gives,

$\hbox{for all}\ n\geq 1,\qquad
g(b_{n-1}-b_{n})\leq(1+\lambda)^{2}\lambda^{2n-2}\alpha^{2}+\lambda^{2n}\alpha^{2}\leq(1+2\lambda)^{2}\lambda^{2n-2}\alpha^{2},$

and so, for all $n\geq 1$,
$\|b_{n-1}-b_{n}\|\leq\sqrt{2}(1+2\lambda)\lambda^{n-1}\alpha$. Adding up this
inequality for $n=1,\dots,m$, we derive that, for all $m\geq 1$,
$\|c-b_{m}\|\leq\sqrt{2}(1+2\lambda)\alpha/(1-\lambda)$. Since
$(1+2\lambda)/(1-\lambda)=1+\varepsilon$, this and (2.7.5) give (2.7.2). Now
set $a=\lim_{n}b_{n}$, so that
$\|c-a\|\leq(1+\varepsilon)\sqrt{2}\sqrt{(f-q)(c)}$. (2.7.5) and the lower
semicontinuity of $f-q$ now imply that $(f-q)(a)\leq 0$, that is to say,
$a\in{\cal P}(f)$. Since $\hbox{\rm dom}\,f\neq\emptyset$, it follows that
${\cal P}(f)\neq\emptyset$ and so, from Lemma 1.9, ${\cal P}(f)$ is a
$q$–positive subset of $B$. We also have

$\hbox{\rm dist}(c,{\cal P}(f))\leq(1+\varepsilon)\sqrt{2}\sqrt{(f-q)(c)},$

and so if we now let $\varepsilon\to 0$, we obtain (2.7.1). This completes the
proof of (a). (b) Suppose first that (i) is satisfied. (2.5.3) implies that
$f\geq q$ on $B$. Let $c\in B$. We choose inductively $b_{1},b_{2},\dots\in B$
such that, for all $n\geq 1$,

$f(b_{n})+g(c-b_{n})+q(c)-\lfloor
c,b_{n}\rfloor=(f-q)(b_{n})+p(c-b_{n})<1/n^{2}.$

Consequently, using (2.1.4) and (2.1.2), for all $n\geq 1$,

$(f-q)(b_{n})<1/n^{2},\ p(c-b_{n})<1/n^{2}$ $None$

and

$f(b_{n})+g(c-b_{n})+q(c)-\|\iota\|\|c\|\|b_{n}\|<1/n^{2}.$ $None$

Since $f\in{\cal PCLSC}(B)$, $f$ dominates a continuous affine function, and
so (2.7.7) and the usual coercivity argument imply that $K:=\sup_{n\geq
1}\|b_{n}\|<\infty$. From (a) and (2.7.6), there exists $a_{n}\in{\cal P}(f)$
such that $\|a_{n}-b_{n}\|\leq\sqrt{2}/n$. Now, from (2.1.5), for all $n\geq
1$,

$\eqalign{|p(c-a_{n})-p(c-b_{n})|&\leq{\textstyle{1\over
2}}(1+\|\iota\|)\|a_{n}-b_{n}\|(2\|c\|+\|a_{n}\|+\|b_{n}\|)\cr&\leq{\textstyle{1\over
2}}(1+\|\iota\|)\big{(}2\|c\|+\big{(}K+\sqrt{2}\big{)}+K\big{)}\sqrt{2}/n.}$

Thus $\lim_{n\to\infty}\big{[}p(c-a_{n})-p(c-b_{n})\big{]}=0$, and (ii)
follows by combining this with (2.7.6). It is trivial that
(ii)$\Longrightarrow$(iii). Suppose, finally, that (iii) is satisfied. Then,
for all $c\in B$,

$\big{(}(f-q)\mathop{\nabla}p\big{)}(c)\leq\inf\nolimits_{a\in{\cal
P}(f)}\big{[}(f-q)(a)+p(c-a)\big{]}=\inf p(c-{\cal P}(f))=0,$

from which $(f-q)\mathop{\nabla}p\leq 0$ on $B$. On the other hand, since
$f-q\geq 0$ on $B$ and, from (2.1.4), $p\geq 0$ on $B$, we have
$(f-q)\mathop{\nabla}p\geq 0$ on $B$. Thus $f$ is a VZ function, giving (i). □

Lemma 2.8. Let $A$ be a closed, $p$–dense and $q$–positive subset of a Banach
SSD space $B$. (a) For all $c\in B$, $\inf q(c-A)\leq 0$ and $\hbox{\rm
dist}(c,A)\leq\sqrt{2}\sqrt{{-}\inf q(c-A)}$. (b) Let $h\in{\cal PC}(B)$,
$h\geq q$ on $B$, and ${\cal P}(h)\supset A$. Then $h$ is a VZ function. (c)
$A$ is a maximally $q$–positive subset of $B$.

Proof. (a) Let $c\in B$. Then $\inf g(c-A)+\inf q(c-A)\leq\inf p(c-A)=0$. Thus
${\textstyle{1\over 2}}\hbox{\rm dist}(c,A)^{2}=\inf g(c-A)\leq-\inf q(c-A)$,
from which (a) is an immediate consequence. (b) Clearly, ${\cal P}(h)$ is also
$p$–dense, and it follows as in Lemma 2.7(b)((iii)$\Longrightarrow$(i)) (which
does not use any semicontinuity) that $h$ is a VZ function, which gives (b).
(c) We suppose that $c\in B$ and $\inf q(c-A)\geq 0$, and we must prove that
$c\in A$. From (a), in fact $\inf q(c-A)=0$ and $\hbox{\rm dist}(c,A)=0$.
Since $A$ is closed, $c\in A$. This completes the proof of (c). □

Theorem 2.9. Let $B$ be a Banach SSD space and $f\in{\cal PCLSC}(B)$ be a VZ
function. Then: (a) For all $c\in B$, $\inf q(c-{\cal P}(f))\leq 0$ and
$\hbox{\rm dist}(c,{\cal P}(f))\leq\sqrt{2}\sqrt{{-}\inf q(c-{\cal P}(f))}$.
(b) Let $h\in{\cal PC}(B)$, $h\geq q$ on $B$, and ${\cal P}(h)\supset{\cal
P}(f)$. Then $h$ is a VZ function. (c) ${\cal P}(f)$ is a maximally
$q$–positive subset of $B$. (d) $f^{@}\in{\cal PCLSC}(B)$, $f^{@}$ is a VZ
function and ${\cal P}\big{(}f^{@}\big{)}={\cal P}(f)$.

Proof. (a), (b) and (c) are immediate from Lemma
2.7(b)((i)$\Longrightarrow$(iii)) and the corresponding parts of Lemma 2.8.
(d) Let $c\in B$. Then, since $q\leq p$ on $B$, Definition 1.10 gives

$q(c)-f^{@}(c)=\inf\nolimits_{b\in B}\big{[}f(b)-\lfloor
b,c\rfloor+q(c)\big{]}=\big{(}(f-q)\mathop{\nabla}q\big{)}(c)\leq\big{(}(f-q)\mathop{\nabla}p\big{)}(c)=0,$

and so $f^{@}\geq q$ on $B$. It now follows from Lemma 1.11(b) that ${\cal
P}\big{(}f^{@}\big{)}\supset{\cal P}(f)$, and so (b) and (c) imply that
$f^{@}$ is a VZ function and ${\cal P}\big{(}f^{@}\big{)}={\cal P}(f)$. Since
${\cal P}(f)\neq\emptyset$, it is evident that $f^{@}\in{\cal PCLSC}(B)$.
(${\cal P}(f)$ is closed because $f$ is lower semicontinuous.) □

Remark 2.10. In general, Theorem 2.9(a) is strictly stronger than Lemma
2.7(a). While this can be proved directly, we will see in Remark 2.17 that it
follows easily from the properties of the $\Phi$–functions. We will also see
in Remark 2.17 that the constant $\sqrt{2}$ in (2.7.1) is sharp.

The proof of Theorem 2.9 relies heavily on the lower semicontinuity of $f$. We
will show in Corollary 2.11 below that part of Theorem 2.9(d) can be recovered
even if $f$ is not assumed to be lower semicontinuous.

Corollary 2.11. Let $B$ be a Banach SSD space and $f\in{\cal PC}(B)$ be a VZ
function. Then $f^{@}\in{\cal PCLSC}(B)$, $f^{@}$ is a VZ function and ${\cal
P}\big{(}f^{@}\big{)}$ is a maximally $q$–positive subset of $B$.

Proof. Let $\overline{f}$ be the lower semicontinuous envelope of $f$. Since
$q$ is continuous and $f\geq q$ on $B$, it follows that $f\geq\overline{f}\geq
q$ on $B$. Thus, from (2.1.4),

$0=(f-q)\mathop{\nabla}p\geq(\overline{f}-q)\mathop{\nabla}p\geq
0\mathop{\nabla}p=0\ \hbox{on}\ B,$

and so $\overline{f}$ is a VZ function. Since $\overline{f}\in{\cal
PCLSC}(B)$, Theorem 2.9(d) implies that $\overline{f}^{@}$ is a VZ function
also. It is well known that $\overline{f}^{*}=f^{*}$ on $B^{*}$ thus,
composing with $\iota$ and using (2.1.6), $\overline{f}^{@}=f^{@}$ on $B$. The
result now follows from Theorem 2.9(d,c), with $f$ replaced by $f^{@}$. □

Definition 2.12. Let $B$ be a Banach SSD space and $A$ be a nonempty
$q$–positive subset of $B$. We define the function $\Theta_{A}\colon\
B^{*}\to\,]{-}\infty,\infty]$ by: for all $b^{*}\in B^{*}$,

$\Theta_{A}(b^{*}):=\sup\nolimits_{a\in A}\big{[}\langle
a,b^{*}\rangle-q(a)\big{]}.$

We define the function $\Phi_{A}\colon\ B\to\,]{-}\infty,\infty]$ by
$\Phi_{A}:=\Theta_{A}\circ\iota$.

We define the function ${}^{*}\Theta_{A}\colon\ B\to\,]{-}\infty,\infty]$ by:
for all $c\in B$,

${}^{*}\Theta_{A}(c):=\sup\nolimits_{b^{*}\in B^{*}}\big{[}\langle
c,b^{*}\rangle-\Theta_{A}(b^{*})\big{]}.$

We collect together in Lemma 2.13 some elementary properties of $\Theta_{A}$,
$\Phi_{A}$, ${}^{*}\Theta_{A}$, and $\Phi_{A}^{@}$. The properties of
$\Phi_{A}$ and $\Phi_{A}^{@}$ have already appeared in [13].

Lemma 2.13. Let $B$ be a Banach SSD space and $A$ be a nonempty $q$–positive
subset of $B$. (a) For all $b\in B$, $\Phi_{A}(b)=\sup\nolimits_{a\in
A}\big{[}\lfloor a,b\rfloor-q(a)\big{]}=q(b)-\inf q(b-A)$. (b)
$\Phi_{A}\in{\cal PCLSC}(B)\quad\hbox{and}\quad\Phi_{A}=q\ \hbox{on}\ A$. (c)
$\Theta_{A}\in{\cal PCLSC}^{*}(B^{*})$. (d) $(^{*}\Theta_{A})^{*}=\Theta_{A}$
and $(^{*}\Theta_{A})^{@}=\Phi_{A}$. (e) ${}^{*}\Theta_{A}\leq q$ on $A$.
Consequently, ${}^{*}\Theta_{A}\in{\cal PCLSC}(B)$. (f)
${}^{*}\Theta_{A}\geq{\Phi_{A}}^{@}\geq\Phi_{A}\vee q\ \hbox{on}\ B$. (g)
${}^{*}\Theta_{A}={\Phi_{A}}^{@}=q\ \hbox{on}\ A$. (h) Let $A$ be maximally
$q$–positive. Then ${}^{*}\Theta_{A}\geq{\Phi_{A}}^{@}\geq\Phi_{A}\geq q$ on
$B$ and $A\subset{\cal P}\big{(}^{*}\Theta_{A}\big{)}$. (i) Let $A$ be
maximally $q$–positive. Then ${\cal P}\big{(}^{*}\Theta_{A}\big{)}={\cal
P}\big{(}{\Phi_{A}}^{@}\big{)}={\cal P}\big{(}\Phi_{A}\big{)}=A$.

Proof. (a) is immediate from (2.1.2), (b) from (a), and (c) from (b) and the
definition of $\Theta_{A}$. The first assertion in (d) follows from (c) and
the Fenchel–Moreau theorem for the locally convex space
$\big{(}B^{*},w(B^{*},B)\big{)}$, while the second assertion follows from the
first by composing with $\iota$, and using (2.1.6) and the definition of
$\Phi_{A}$. (e) Let $a\in A$. The definition of $\Theta_{A}$ implies that, for
all $b^{*}\in B^{*}$, $\langle a,b^{*}\rangle-\Theta_{A}(b^{*})\leq q(a)$.
Taking the supremum over $b^{*}\in B^{*}$, ${}^{*}\Theta_{A}(a)\leq q(a)$, as
required. (f) Let $c\in B$. Then, from (2.1.2), the definition of $\Phi_{A}$
and (b),

$\eqalignno{{}^{*}\Theta_{A}(c)&\geq\sup\nolimits_{b\in B}\big{[}\langle
c,\iota(b)\rangle-\Theta_{A}(\iota(b))\big{]}\cr&=\sup\nolimits_{b\in
B}\big{[}\lfloor
c,b\rfloor-\Phi_{A}(b)\big{]}\quad\big{(}={\Phi_{A}}^{@}(c)\big{)}\cr&\geq\big{[}\lfloor
c,c\rfloor-\Phi_{A}(c)\big{]}\vee\sup\nolimits_{a\in A}\big{[}\lfloor
c,a\rfloor-\Phi_{A}(a)\big{]}\cr&=\big{[}2q(c)-\Phi_{A}(c)\big{]}\vee\sup\nolimits_{a\in
A}\big{[}\lfloor
c,a\rfloor-q(a)\big{]}\cr&=\big{[}2q(c)-\Phi_{A}(c)\big{]}\vee\Phi_{A}(c).}$

Now if $\Phi_{A}(c)=\infty$ then obviously
$\big{[}2q(c)-\Phi_{A}(c)\big{]}\vee\Phi_{A}(c)\geq q(c)$, while if
$\Phi_{A}(c)\in\hbox{\tenmsb R}$ then
$\big{[}2q(c)-\Phi_{A}(c)\big{]}\vee\Phi_{A}(c)\geq{\textstyle{1\over
2}}[2q(c)-\Phi_{A}(c)\big{]}+{\textstyle{1\over 2}}\Phi_{A}(c)=q(c)$. Thus
${\Phi_{A}}^{@}(c)\geq\Phi(c)\vee q(c)$. This completes the proof of (f). (g)
is immediate from (e) and (f). (h) In this case, for all $b\in B\setminus A$,
there exists $a\in A$ such that $q(b-a)<0$, and so $\inf q(b-A)<0$. Thus, from
(a), $\Phi_{A}>q$ on $B\setminus A$. Combining this with (b), $\Phi_{A}\geq q$
on $B$ and ${\cal P}\big{(}\Phi_{A}\big{)}=A$. Thus (h) follows from (f) and
(g). It is clear from (h) that $A\subset{\cal
P}\big{(}^{*}\Theta_{A}\big{)}\subset{\cal
P}\big{(}{\Phi_{A}}^{@}\big{)}\subset{\cal P}\big{(}\Phi_{A}\big{)}$, and so
(i) follows from the maximality of $A$. □

Remark 2.14. We will see in (2.15.2) and (2.15.3) that ${}^{*}\Theta$ and
${\Phi_{A}}^{@}$ are both “upper limiting” functions in various situations, so
the question arises whether these two functions are identical. If
${}^{*}\Theta_{A}={\Phi_{A}}^{@}$ then ${}^{*}\Theta_{A}$ is obviously
$w(B,B)$–lower semicontinuous. If, conversely, ${}^{*}\Theta_{A}$ is
$w(B,B)$–lower semicontinuous then the Fenchel–Moreautheorem for the (possibly
nonhausdorff) locally convex space $\big{(}B,w(B,B)\big{)}$ and Lemma 2.13(d)
imply that
${}^{*}\Theta_{A}=\big{(}^{*}\Theta_{A}\big{)}^{@@}={\Phi_{A}}^{@}$. The
author is grateful to Constantin Zălinescu for the following example showing
that, in general, the functions ${}^{*}\Theta$ and ${\Phi_{A}}^{@}$ are not
identical. Let $B$ be a Banach space, $\lfloor\cdot,\cdot\rfloor=0$ on
$B\times B$ and $A$ be a nonempty proper closed convex subset of $B$. Then
${}^{*}\Theta_{A}$ is the indicator function of $A$ and ${\Phi_{A}}^{@}=0$ on
$B$. We do not know what the situation is if $A$ is maximally $q$–positive, or
in the special situation of Example 2.4. For the convenience of the reader, we
will give a proof of the Fenchel–Moreau theorem for nonhausdorff locally
convex spaces in Theorem 6.1.

Theorem 2.15. Let $B$ be a Banach SSD space.

(a) Let $f\in{\cal PCLSC}(B)$, $f\geq q$ on $B$ and $A:={\cal
P}(f)\neq\emptyset$. Then ${}^{*}\Theta_{A}\geq f\geq\Phi_{A}$ on $B$ and
${\Phi_{A}}^{*}\geq f^{*}\geq\Theta_{A}$ on $B^{*}$. (b) Let $A$ be a
maximally $q$–positive subset of $B$, $h\in{\cal PC}(B)$ and
${}^{*}\Theta_{A}\geq h\geq\Phi_{A}$ on $B$. Then $h\geq q$ on $B$, $h^{@}\geq
q$ on $B$ and ${\cal P}(h)={\cal P}\big{(}h^{@}\big{)}=A$. (c) Let $f\in{\cal
PCLSC}(B)$ be a VZ function and $A:={\cal P}(f)$. Then

${}^{*}\Theta_{A}\geq f\geq\Phi_{A}\geq q\ \hbox{on}\
B\quad\hbox{and}\quad{\Phi_{A}}^{*}\geq f^{*}\geq\Theta_{A}\ \hbox{on}\
B^{*}.$ $None$

Now let $h\in{\cal PC}(B)$ and ${}^{*}\Theta_{A}\geq h\geq\Phi_{A}$ on $B$.
Then $h$ and $h^{@}$ are VZ functions. In particular, ${\cal
P}\big{(}^{*}\Theta_{A}\big{)}={\cal P}\big{(}{\Phi_{A}}^{@}\big{)}={\cal
P}\big{(}\Phi_{A}\big{)}={\cal P}(f)$ and $\Phi_{A}$, ${\Phi_{A}}^{@}$ and
${}^{*}\Theta_{A}$ are all VZ functions.

Proof. (a) Let $b\in B$ and $a\in{\cal P}(f)$. Then, from Lemma 1.11(a),
$f(b)\geq\lfloor b,a\rfloor-q(a)$. Taking the supremum over $a\in{\cal P}(f)$
and using Lemma 2.13(a), $f(b)\geq\Phi_{A}(b)$. Thus $f\geq\Phi_{A}$ on $B$
and, taking conjugates, ${\Phi_{A}}^{*}\geq f^{*}$ on $B^{*}$. Now, for all
$b^{*}\in B^{*}$,

$\eqalign{f^{*}(b^{*})&=\sup\nolimits_{b\in B}\big{[}\langle
b,b^{*}\rangle-f(b)\big{]}\geq\sup\nolimits_{a\in{\cal P}(f)}\big{[}\langle
a,b^{*}\rangle-f(a)\big{]}\cr&=\sup\nolimits_{a\in{\cal P}(f)}\big{[}\langle
a,b^{*}\rangle-q(a)\big{]}=\Theta_{A}(b^{*}).}$

Thus $f^{*}\geq\Theta_{A}$ on $B^{*}$. Taking conjugates and using the
Fenchel–Moreau theorem for the normed space $B$, ${}^{*}\Theta_{A}\geq f$ on
$B$. This completes the proof of (a). (b) From Lemma 2.13(h),

${}^{*}\Theta_{A}\geq h\geq\Phi_{A}\geq q\ \hbox{on}\ B,\quad\hbox{from
which}\quad{\cal P}\big{(}^{*}\Theta_{A}\big{)}\subset{\cal P}(h)\subset{\cal
P}\big{(}\Phi_{A}\big{)}.$ $None$

It is clear from our assumptions that ${\Phi_{A}}^{@}\geq
h^{@}\geq(^{*}\Theta_{A})^{@}$ on $B$. If we now combine this with Lemma
2.13(d,h), we derive that

${\Phi_{A}}^{@}\geq h^{@}\geq\Phi_{A}\geq q\ \hbox{on}\ B,\quad\hbox{from
which}\quad{\cal P}\big{(}{\Phi_{A}}^{@}\big{)}\subset{\cal
P}\big{(}h^{@}\big{)}\subset{\cal P}\big{(}\Phi_{A}\big{)}.$ $None$

(b) now follows from (2.15.2), (2.15.3) and Lemma 2.13(i). (c) The assertions
about $f$ follow from (2.5.3), Theorem 2.9(c), (a) and Lemma 2.13(h), the
assertions about $h$ and $h^{@}$ follow from Theorem 2.9(c,b) and (b), and
then the assertions about $\Phi_{A}$, ${\Phi_{A}}^{@}$ and ${}^{*}\Theta_{A}$
follow from Theorem 2.9(c) and Lemma 2.13(h,i). □

In Theorem 2.16 below, we show that ${}^{*}\Theta_{A}$ has a certain maximal
property. This result was motivated by results originally proved by Burachik
and Svaiter in [1] for maximally monotone multifunctions.

Theorem 2.16. Let $A$ be a nonempty $q$–positive subset of a Banach SSD space
$B$ and

$\sigma_{A}:=\sup\big{\\{}h\colon\ h\in{\cal PCLSC}(B),\ h\leq q\ \hbox{on}\
A\big{\\}}.$

Then ${}^{*}\Theta_{A}=\sigma_{A}$ on $B$.

Proof. Let $h\in{\cal PCLSC}(B)$ and $h\leq q$ on $A$. The Fenchel Young
inequality implies that, for all $b^{*}\in B^{*}$ and $a\in A$,
$h^{*}(b^{*})\geq\langle a,b^{*}\rangle-h(a)\geq\langle a,b^{*}\rangle-q(a)$.
Thus, taking the supremum over $a\in A$, $h^{*}(b^{*})\geq\Theta_{A}(b^{*})$.
In other words, $h^{*}\geq\Theta_{A}$ on $B^{*}$. Taking conjugates and using
the Fenchel Moreau theorem for the normed space $B$, ${}^{*}\Theta_{A}\geq h$
on $B$. It follows by taking the supremum over $h$ that
${}^{*}\Theta_{A}\geq\sigma_{A}$ on $B$. On the other hand, it is clear from
Lemma 2.13(e) that $\sigma_{A}\geq{{}^{*}\Theta_{A}}\ \hbox{on}\ B$. □

Remark 2.17. Let $B$ be a Banach SSD space and $f\in{\cal PCLSC}(B)$ be a VZ
function. We know from Theorem 2.15(c) that ${\cal P}\big{(}\Phi_{{\cal
P}(f)}\big{)}={\cal P}(f)$, $\Phi_{{\cal P}(f)}$ is a VZ function and
$\Phi_{{\cal P}(f)}\leq f$ on $B$. Thus Lemma 2.7(a) implies that, for all
$c\in B$,

$\hbox{\rm dist}(c,{\cal P}(f))=\hbox{\rm dist}\big{(}c,{\cal
P}\big{(}\Phi_{{\cal P}(f)}\big{)}\big{)}\leq\sqrt{2}\sqrt{\big{(}\Phi_{{\cal
P}(f)}-q\big{)}(c)}\leq\sqrt{2}\sqrt{(f-q)(c)}.$

From Lemma 2.13(a), $\big{(}\Phi_{{\cal P}(f)}-q\big{)}(c)=\Phi_{{\cal
P}(f)}(c)-q(c)=-\inf q(c-{\cal P}(f))$, thus we have

$\hbox{\rm dist}(c,{\cal P}(f))\leq\sqrt{2}\sqrt{-\inf q(c-{\cal
P}(f))}\leq\sqrt{2}\sqrt{(f-q)(c)}.$

This shows that Theorem 2.9(a) is as least as strong as Lemma 2.7(a). Now let
$E:=\hbox{\tenmsb R}$ and $B$ be the Banach SSD space $\hbox{\tenmsb R}^{2}$
as in Example 2.4, using the norm $\|\cdot\|_{2,1}$. Define $f\in{\cal
PCLSC}(B)$ by $f(x_{1},x_{2}):={\textstyle{1\over 2}}(x_{1}^{2}+x_{2}^{2})$.
Then $(f-q)(x_{1},x_{2})={\textstyle{1\over
2}}(x_{1}^{2}+x_{2}^{2})-x_{1}x_{2}={\textstyle{1\over 2}}(x_{1}-x_{2})^{2}$
and $p(x_{1},x_{2})={\textstyle{1\over
2}}(x_{1}^{2}+x_{2}^{2})+x_{1}x_{2}={\textstyle{1\over 2}}(x_{1}+x_{2})^{2}$.
Let $c:=(z_{1},z_{2})\in B$ and $b:=\big{(}{\textstyle{1\over
2}}(z_{1}+z_{2}),{\textstyle{1\over 2}}(z_{1}+z_{2})\big{)}\in B$. Then
$(f-q)(b)=0$ and $p(c-b)=0$. Consequently, $f$ is a VZ function. Now ${\cal
P}(f)$ is the diagonal of $\hbox{\tenmsb R}^{2}$ and so, by direct
computation, for all $c=(x_{1},x_{2})\in\hbox{\tenmsb R}^{2}$, $-\inf
q(c-{\cal P}(f))=\textstyle{1\over 4}(x_{1}-x_{2})^{2}$. Since
$\textstyle{1\over 4}(x_{1}-x_{2})^{2}<{\textstyle{1\over
2}}(x_{1}-x_{2})^{2}$ when $x_{1}\neq x_{2}$, Theorem 2.9(a) is strictly
stronger than Lemma 2.7(a) in this case.

Now let $h:=\Phi_{{\cal P}(f)}$. Lemma 2.13(a) gives us that, for all
$(x_{1},x_{2})\in B$,

$\sqrt{(h-q)(x_{1},x_{2})}=\sqrt{\textstyle{1\over
4}(x_{1}-x_{2})^{2}}={\textstyle{1\over 2}}|x_{1}-x_{2}|.$

On the other hand, by direct computation, $\hbox{\rm
dist}\big{(}(x_{1},x_{2}),{\cal
P}(h)\big{)}=\textstyle{1\over\sqrt{2}}|x_{1}-x_{2}|$. Thus the constant
$\sqrt{2}$ in (2.7.1) is sharp. The genesis of this argument and example can
be found in the results of Martínez-Legaz and Théra in [6].

Remark 2.18. We note that the inequalities for $B$ in (2.15.1) have four
functions, while the inequality for $B^{*}$ has only three. The reason for
this is that we do not have a function on $B^{*}$ that plays the role that the
function $q$ plays on $B$. We will introduce such a function in Definition
4.1.

3 Applications of Section 2 to $E\times E^{*}$

In this section, we suppose that $E$ is a nonzero Banach space, and follow the
notation of Example 2.3. Let $A$ be a nonempty monotone subset of $E\times
E^{*}$. In this case, the definitions and results obtained in Definition 2.12
and Lemma 2.13 specialize as follows. The function $\Theta_{A}\in{\cal
PCLSC}^{*}(E^{**}\times E^{*})$ is defined by:

$\Theta_{A}(x^{**},x^{*}):=\sup\nolimits_{(s,s^{*})\in A}\big{[}\langle
s,x^{*}\rangle+\langle s^{*},x^{**}\rangle-\langle s,s^{*}\rangle\big{]}.$

The function $\Phi_{A}\in{\cal PCLSC}(E\times E^{*})$ is defined by:

$\Phi_{A}(x,x^{*})=\sup\nolimits_{(s,s^{*})\in A}\big{[}\langle
x,s^{*}\rangle+\langle s,x^{*}\rangle-\langle s,s^{*}\rangle\big{]}.$

$\Phi_{A}$ is the Fitzpatrick function of $A$, first introduced in [2], which
has been discussed by many authors in recent years. The function
${}^{*}\Theta_{A}\in{\cal PCLSC}(E\times E^{*})$ is defined by:

${}^{*}\Theta_{A}(y,y^{*}):=\sup\nolimits_{(x^{**},x^{*})\in E^{**}\times
E^{*}}\big{[}\langle y,x^{*}\rangle+\langle
y^{*},x^{**}\rangle-\Theta_{A}(x^{**},x^{*})\big{]}.$

Then $(^{*}\Theta_{A})^{*}=\Theta_{A}$ and $(^{*}\Theta_{A})^{@}=\Phi_{A}$.
Furthermore,

${}^{*}\Theta_{A}\geq{\Phi_{A}}^{@}\geq\Phi_{A}\vee q\ \hbox{on}\ E\times
E^{*}\quad\hbox{and}\quad^{*}\Theta_{A}={\Phi_{A}}^{@}=\Phi_{A}=q\ \hbox{on}\
A.$

If $f\in{\cal PC}(E\times E^{*})$ and $f\geq q$ on $E\times E^{*}$ then we
define ${\cal M}f$ to be the monotone set $\\{(x,x^{*})\in E\times E^{*}\colon
f(x,x^{*})=\langle x,x^{*}\rangle\\}$. ${\cal M}f$ is identical with ${\cal
P}(f)$ as in Definition 1.8, but the “${\cal M}$” notation seems more
appropriate in this case. Continuing with the consequences of Lemma 2.13, we
have:

${}^{*}\Theta_{A}\geq{\Phi_{A}}^{@}\geq\Phi_{A}\geq q\ \hbox{on}\ E\times
E^{*}\quad\hbox{and}\quad{\cal M}\big{(}^{*}\Theta_{A}\big{)}={\cal
M}\big{(}{\Phi_{A}}^{@}\big{)}={\cal M}\big{(}\Phi_{A}\big{)}=A.$

The following results are then immediate from Theorems 2.15 and 2.16. The
expression $\sup\big{\\{}h\colon\ h\in{\cal PCLSC}(E\times E^{*}),\ h\leq q\
\hbox{on}\ A\big{\\}}$ that appears in Theorem 3.1(b) was first introduced by
Burachik and Svaiter in [1] (for $A$ maximally monotone) and further studied
by Marques Alves and Svaiter in [5]. The analysis of Lemma 2.13 and Theorem
2.15 suggests that the natural framework in which to consider these results is
that of Banach SSD spaces.

Theorem 3.1. Let $E$ be a nonzero Banach space, $E\times E^{*}$ be normed as
in Example 2.3, and $A$ be a nonempty monotone subset of $E\times E^{*}$.

(a) Let $f\in{\cal PCLSC}(E\times E^{*})$, $f\geq q$ on $E\times E^{*}$ and
$A:={\cal M}f\neq\emptyset$. Then

${}^{*}\Theta_{A}\geq f\geq\Phi_{A}\ \hbox{on}\ E\times
E^{*},\quad\hbox{and}\quad{\Phi_{A}}^{*}\geq f^{*}\geq\Theta_{A}\ \hbox{on}\
E^{**}\times E^{*}.$

(b) Let $A$ be maximally monotone, $h\in{\cal PC}(E\times E^{*})$ and
${}^{*}\Theta_{A}\geq h\geq\Phi_{A}$ on $E\times E^{*}$. Then $h\geq q$ on
$E\times E^{*}$, $h^{@}\geq q$ on $E\times E^{*}$ and ${\cal M}h={\cal
M}\big{(}h^{@}\big{)}=A$. (c) Let $f\in{\cal PCLSC}(E\times E^{*})$ be a VZ
function and $A:={\cal M}f$. Then

${}^{*}\Theta_{A}\geq f\geq\Phi_{A}\geq q\ \hbox{on}\ E\times
E^{*}\quad\hbox{and}\quad{\Phi_{A}}^{*}\geq f^{*}\geq\Theta_{A}\ \hbox{on}\
E^{**}\times E^{*}$

Now let $h\in{\cal PC}(E\times E^{*})$ and ${}^{*}\Theta_{A}\geq
h\geq\Phi_{A}$ on $E\times E^{*}$. Then $h$ and $h^{@}$ are VZ functions. In
particular, ${\cal M}\big{(}^{*}\Theta_{A}\big{)}={\cal
M}\big{(}{\Phi_{A}}^{@}\big{)}={\cal M}\big{(}\Phi_{A}\big{)}=A$, and
$\Phi_{A}$, ${\Phi_{A}}^{@}$ and ${}^{*}\Theta_{A}$ are all VZ functions.

${}^{*}\Theta_{A}=\sup\big{\\{}h\colon\ h\in{\cal PCLSC}(E\times E^{*}),\
h\leq q\ \hbox{on}\ A\big{\\}}.$ $None$

4 Banach SSD dual spaces

Definition 4.1. Let $(B,\|\cdot\|)$ be a Banach SSD space and
$(B^{*},\|\cdot\|)$ be the norm–dual of $B$. We say that
$(B^{*},\lceil\cdot,\cdot\rceil)$ is a Banach SSD dual of $B$ if
$\lceil\cdot,\cdot\rceil\colon\ B^{*}\times B^{*}\to\hbox{\tenmsb R}$ is a
symmetric bilinear form,

$\hbox{for all}\ b\in B\ \hbox{and}\ c^{*}\in
B^{*},\qquad\lceil\iota(b),c^{*}\rceil=\langle b,c^{*}\rangle.$ $None$

Writing $\widetilde{q}(c^{*}):={\textstyle{1\over 2}}\lceil c^{*},c^{*}\rceil$
and $\widetilde{p}(c^{*}):={\textstyle{1\over
2}}\|c^{*}\|^{2}+\widetilde{q}(c^{*})$, we suppose also that

$\widetilde{p}\geq 0\ \hbox{on}\ B^{*}.$ $None$

Now if we take $c^{*}=\iota(c)$ in (4.1.1) and use (2.1.2), we obtain

$\hbox{for all}\ b,c\in
B,\qquad\lceil\iota(b),\iota(c)\rceil=\big{\langle}b,\iota(c)\big{\rangle}=\lfloor
b,c\rfloor,$ $None$

from which

$\widetilde{q}\circ\iota=q.$ $None$

It is easy to see from these definitions that,

$\hbox{for all}\ b^{*}\in
B^{*},\qquad\Theta_{A}(b^{*})=\widetilde{q}(b^{*})-\inf\widetilde{q}(b^{*}-\iota(A)).$
$None$

This should be compared with Lemma 2.13(a).

Definition 4.2. Let $(B,\|\cdot\|)$ be a Banach SSD space and
$(B^{*},\lceil\cdot,\cdot\rceil)$ be a Banach SSD dual of $B$. We say that
$\iota(B)$ is $\widetilde{p}$–dense in $B^{*}$ if

$\hbox{for all}\ b^{*}\in
B^{*},\quad\inf\widetilde{p}\big{(}b^{*}-\iota(B)\big{)}=0.$ $None$

Remark 4.3. In Example 1.3 with $\|T\|\leq 1$ (see also Remark 2.2), for all
$c\in B$, $\iota(c)=Tc$. Suppose now that $T^{2}$ is the identity on $B$.
Since $B^{*}=B$,

$\hbox{for all}\ b\in B\ \hbox{and}\ c^{*}\in
B^{*}=B,\qquad\lfloor\iota(b),c^{*}\rfloor=\lfloor Tb,c^{*}\rfloor=\langle
T^{2}b,c^{*}\rangle=\langle b,c^{*}\rangle.$

Thus (4.1.1) is satisfied with
$\lceil\cdot,\cdot\rceil:=\lfloor\cdot,\cdot\rfloor$, and so
($B,\lfloor\cdot,\cdot\rfloor$) is its own Banach SSD dual. We note that
$T^{2}$ is the identity on $B$ in (a), (b) and (c) of Example 1.3.

Example 4.4. We now continue our discussion of Examples 1.4, 2.3 and 2.4. We
recall that $B=E\times E^{*}$, $B^{*}=E^{**}\times E^{*}$ and, for all
$(x,x^{*})\in B$, $\iota(x,x^{*})=(\widehat{x},x^{*})$. We define the
symmetric bilinear form $\lceil\cdot,\cdot\rceil\colon\ B^{*}\times
B^{*}\to\hbox{\tenmsb R}$ by

$\lceil b^{*},c^{*}\rceil:=\langle y^{*},x^{**}\rangle+\langle
x^{*},y^{**}\rangle\quad\big{(}b^{*}=(x^{**},x^{*})\in B^{*},\
c^{*}=(y^{**},y^{*})\in B^{*}\big{)}.$

It is then easily checked from (2.3.1) that (4.1.1) is satisfied and, for all
$c^{*}=(y^{**},y^{*})\in B^{*}$, $\widetilde{q}(c^{*})={\textstyle{1\over
2}}\big{[}\langle y^{*},y^{**}\rangle+\langle
y^{*},y^{**}\rangle\big{]}=\langle y^{*},y^{**}\rangle$. We now discuss
briefly the limitations of this definition. Let $E:=\hbox{\tenmsb R}$ and $B$
be the SSD space $\hbox{\tenmsb R}^{2}$ as in Example 1.4, using the norm
$2\|\cdot\|_{2,1}$. As we observed in Example 2.4, $\hbox{\tenmsb R}^{2}$ is a
Banach SSD space under $\|\cdot\|_{2,1}$, and consequently also a Banach SSD
space under the larger norm $2\|\cdot\|_{2,1}$. Since $\iota$ is the identity
on $\hbox{\tenmsb R}^{2}$, (4.1.3) implies that
$\lceil\cdot,\cdot\rceil:=\lfloor\cdot,\cdot\rfloor$. Now the norm on
$B^{*}=B$ dual to $2\|\cdot\|_{2,1}$ is ${\textstyle{1\over
2}}\|\cdot\|_{2,1}$. Since $\widetilde{p}(1,-1)={1\over 8}(2)+(1)(-1)=-{3\over
4}<0$, $B$ does not admit a Banach SSD dual. We now return to the general
case. If $c^{*}=(y^{**},y^{*})\in B^{*}$ then

${\textstyle{1\over
2}}\|c^{*}\|_{1,\tau}^{2}+\widetilde{q}(c^{*})\geq\textstyle{1\over
4}\big{(}\tau\|y^{**}\|+\|y^{*}\|/\tau\big{)}^{2}-\|y^{**}\|\|y^{*}\|=\textstyle{1\over
4}\big{(}\tau\|y^{**}\|-\|y^{*}\|/\tau\big{)}^{2}\geq 0.$

Consequently, $\big{(}B^{*},\|\cdot\|_{1,\tau}\big{)}$ is a Banach SSD dual of
$\big{(}B,\|\cdot\|_{\infty,\tau}\big{)}$. Since
$\|\cdot\|_{\infty,\tau}\geq\|\cdot\|_{2,\tau}\geq\|\cdot\|_{1,\tau}$ on
$B^{*}$, $\big{(}B^{*},\|\cdot\|_{2,\tau}\big{)}$ is a Banach SSD dual of
$\big{(}B,\|\cdot\|_{2,\tau}\big{)}$ and
$\big{(}B^{*},\|\cdot\|_{\infty,\tau}\big{)}$ is a Banach SSD dual of
$\big{(}B,\|\cdot\|_{1,\tau}\big{)}$. Next, if $b^{*}=(y^{**},y^{*})\in B^{*}$
and $\varepsilon>0$ then there exists $z^{*}\in E^{*}$ such that
$\|z^{*}\|\leq\|\tau y^{**}\|$ and $\langle z^{*},\tau y^{**}\rangle\geq\|\tau
y^{**}\|^{2}-\varepsilon$. Let $c:=(0,y^{*}+\tau z^{*})\in B$, so that
$b^{*}-\iota(c)=(y^{**},-\tau z^{*})\in B^{*}$. Thus

$\eqalign{{\textstyle{1\over
2}}\|b^{*}-\iota(c)\|_{\infty,\tau}^{2}&+\widetilde{q}(b^{*}-\iota(c))=\big{(}\tau\|y^{**}\|\vee\|z^{*}\|\big{)}^{2}-\langle\tau
z^{*},y^{**}\rangle\cr&=\big{(}\|\tau y^{**}\|\vee\|z^{*}\|\big{)}^{2}-\langle
z^{*},\tau y^{**}\rangle=\|\tau y^{**}\|^{2}-\langle z^{*},\tau
y^{**}\rangle\leq\varepsilon.}$

Consequently, if $B$ is normed by $\|\cdot\|_{1,\tau}$ then $\iota(B)$ is
$\widetilde{p}$–dense in $B^{*}$. Since
$\|\cdot\|_{1,\tau}\leq\|\cdot\|_{2,\tau}\leq\|\cdot\|_{\infty,\tau}$ on
$B^{*}$, the same is true if $B$ is normed by $\|\cdot\|_{2,\tau}$ or
$\|\cdot\|_{\infty,\tau}$.

We now recall Rockafellar’s formula for the conjugate of a sum:

Lemma 4.5. Let $X$ be a nonzero real Banach space and $f\in{\cal PC}(X)$, and
let $h\in{\cal PC}(X)$ be real–valued and continuous. Then, for all $x^{*}\in
X^{*}$,

$(f+h)^{*}(x^{*})=\min\nolimits_{y^{*}\in
X^{*}}\big{[}f^{*}(y^{*})+h^{*}(x^{*}-y^{*})\big{]}.$

Proof. See Rockafellar, [8, Theorem 3(a), p. 85], Zălinescu, [16, Theorem
2.8.7(iii), p. 127], or [13, Corollary 10.3, p. 52]. □

Remark 4.6. [13, Theorem 7.4, p. 43] contains a version of the Fenchel duality
theorem with a sharp lower bound on the functional obtained.

Our next result exhibits a certain pleasing symmetry between $B$ and $B^{*}$.

Lemma 4.7. Let $B$ be a Banach SSD space with a Banach SSD dual $B^{*}$ and
$f\in{\cal PC}(B)$. Then
$\big{(}(f-q)\mathop{\nabla}p\big{)}+\big{(}(f^{*}-\widetilde{q})\mathop{\nabla}\widetilde{p}\big{)}\circ\iota=0$
on $B$.

Proof. Let $c\in B$. Define $h\colon\ B\to\hbox{\tenmsb R}$ by $h(b):=g(c-b)$.
Then, by direct computation using the fact that $g$ is an even function,

$\hbox{for all}\ c^{*}\in B^{*},\qquad h^{*}(c^{*})=g^{*}(c^{*})+\langle
c,c^{*}\rangle.$ $None$

Then, using (2.1.2), the continuity of $h$, Lemma 4.5, (4.7.1), (4.1.4) and
the fact that, for all $c^{*}\in B^{*}$, $g^{*}(c^{*})={\textstyle{1\over
2}}\|c^{*}\|^{2}$,

$\eqalign{-\big{(}(f-q)\mathop{\nabla}p\big{)}(c)&=\sup\nolimits_{b\in
B}\big{[}-(f-q)(b)-p(c-b)\big{]}\cr&=\sup\nolimits_{b\in B}\big{[}\langle
b,\iota(c)\rangle-f(b)-h(b)\big{]}-q(c)=(f+h)^{*}\big{(}\iota(c)\big{)}-q(c)\cr&=\min\nolimits_{b^{*}\in
B^{*}}\big{[}f^{*}(b^{*})+h^{*}\big{(}\iota(c)-b^{*}\big{)}\big{]}-q(c)\cr&=\min\nolimits_{b^{*}\in
B^{*}}\big{[}f^{*}(b^{*})+g^{*}\big{(}\iota(c)-b^{*}\big{)}+\big{\langle}c,\iota(c)-b^{*}\big{\rangle}\big{]}-q(c)\cr&=\min\nolimits_{b^{*}\in
B^{*}}\big{[}f^{*}(b^{*})+g^{*}\big{(}\iota(c)-b^{*}\big{)}-\lceil\iota(c),b^{*}\rceil+\widetilde{q}\big{(}\iota(c)\big{)}\big{]}\cr&=\min\nolimits_{b^{*}\in
B^{*}}\big{[}(f^{*}-\widetilde{q})(b^{*})+\widetilde{p}\big{(}\iota(c)-b^{*}\big{)}\big{]}\cr&=\big{(}(f^{*}-\widetilde{q})\mathop{\nabla}\widetilde{p}\big{)}\big{(}\iota(c)\big{)}.}$

This completes the proof of Lemma 4.7. □

Definition 4.8. Let $B$ be a Banach SSD space with Banach SSD dual $B^{*}$ and
$f\in{\cal PC}(B)$. We say that $f$ is an MAS function if $f\geq q$ on $B$ and
$f^{*}\geq\widetilde{q}$ on $B^{*}$. This is an extension to Banach SSD spaces
of the concept introduced by Marques Alves and Svaiter in [4] for the
situation described in Example 4.4.

Theorem 4.9. Let $B$ be a Banach SSD space with Banach SSD dual $B^{*}$ and
$f\in{\cal PC}(B)$.

(a) Let $f$ be an MAS function. Then $f$ is a VZ function.

(b) Let $\iota(B)$ be $\widetilde{p}$–dense in $B^{*}$ and $f$ be a VZ
function. Then $f$ is an MAS function.

(c) Let $\iota(B)$ be $\widetilde{p}$–dense in $B^{*}$. Then $f$ is a VZ
function if, and only if, $f$ is an MAS function.

Proof. (a) We have (using (2.1.4)) $f-q\geq 0$ and $p\geq 0$ on $B$, and
(using (4.1.2)), $f^{*}-\widetilde{q}\geq 0$ and $\widetilde{p}\geq 0$ on
$B^{*}$. Thus $(f-q)\mathop{\nabla}p\geq 0$ and
$\big{(}(f^{*}-\widetilde{q})\mathop{\nabla}\widetilde{p}\big{)}\circ\iota\geq
0$ on $B$. It now follows from Lemma 4.7 that $f$ is a VZ function.

(b) Let $b^{*}\in B^{*}$ and $c\in B$. Then, from Lemma 4.7 again,

$(f^{*}-\widetilde{q})(b^{*})+\widetilde{p}\big{(}\iota(c)-b^{*}\big{)}\geq\big{(}(f^{*}-\widetilde{q})\mathop{\nabla}\widetilde{p}\big{)}\big{(}\iota(c)\big{)}=-\big{(}(f-q)\mathop{\nabla}p\big{)}(c)=0.$

Taking the infimum over $c\in B$ and using (4.2.1),
$(f^{*}-\widetilde{q})(b^{*})\geq 0$ on $B^{*}$. Since this holds for all
$b^{*}\in B^{*}$, $f$ is an MAS function.

(c) is immediate from (a) and (b). □

In Theorem 4.10, we shift the emphasis from the properties of a given function
$f\in{\cal PC}(B)$ to the properties of a given maximally $q$–positive subset
$A$ of $B$. We note that (a), (b), (c), (f) and (g) of Theorem 4.10 do not
involve any functions on $B$ other than those introduced in Definition 2.12.

Theorem 4.10. Let $(B,\|\cdot\|)$ be a Banach SSD space with Banach SSD dual
$B^{*}$ and $\iota(B)$ be $\widetilde{p}$–dense in $B^{*}$. Let $A$ be a
maximally $q$–positive subset of $B$. Then the following conditions are
equivalent:

(a) For all $b^{*}\in B^{*}$,
$\inf\widetilde{q}\big{(}b^{*}-\iota(A)\big{)}\leq 0$. (b)
$\Theta_{A}\geq\widetilde{q}$ on $B^{*}$. (c)
${\Phi_{A}}^{*}\geq\widetilde{q}$ on $B^{*}$. (d) There exists an MAS function
$f\in{\cal PCLSC}(B)$ such that ${\cal P}(f)=A$. (e) There exists a VZ
function $f\in{\cal PCLSC}(B)$ such that ${\cal P}(f)=A$. (f) $\Phi_{A}$ is a
VZ function. (g) ${}^{*}\Theta_{A}$ is a VZ function. (b1) If $h\in{\cal
PC}(B)$ and ${}^{*}\Theta_{A}\geq h$ on $B$ then $h^{*}\geq\widetilde{q}$ on
$B^{*}$. (b2) If $h\in{\cal PCLSC}(B)$ and ${}^{*}\Theta_{A}\geq
h\geq\Phi_{A}$ on $B$ then $h^{*}\geq\widetilde{q}$ on $B^{*}$. (c1) There
exists $h\in{\cal PCLSC}(B)$ such that ${}^{*}\Theta_{A}\geq h\geq\Phi_{A}$ on
$B$ and $h^{*}\geq\widetilde{q}$ on $B^{*}$. (c2) There exists $h\in{\cal
PC}(B)$ such that $h\geq\Phi_{A}$ on $B$ and $h^{*}\geq\widetilde{q}$ on
$B^{*}$.

Proof. The equivalence of (a) and (b) is immediate from (4.1.5). Taking the
conjugate of the inequality in Lemma 2.13(f) and using Lemma 2.13(d) implies
that ${\Phi_{A}}^{*}\geq\Theta_{A}$ on $B^{*}$. Thus (b)$\Longrightarrow$(c).
If (c) is satisfied then Lemma 2.13(b,h,i) give (d) with $f:=\Phi_{A}$. It is
immediate from Theorem 4.9(c) that (d)$\Longrightarrow$(e). If (e) is
satisfied then Theorem 2.15(c) gives (f) and (g). If (f) or (g) is satisfied
then, from Theorem 4.9(c) again, $\Phi_{A}$ or ${}^{*}\Theta_{A}$
(respectively) are MAS functions. The first of these possibilities implies
(c), and the second of these possibilities together with Lemma 2.13(d) implies
(b). Thus (a), (b), (c), (d), (e), (f) and (g) are equivalent.

If $h\in{\cal PC}(B)$ and ${}^{*}\Theta_{A}\geq h$ on $B$ then, from Lemma
2.13(d), $h^{*}\geq(^{*}\Theta_{A})^{*}=\Theta_{A}$ on $B^{*}$, thus (b)
implies (b1). It is trivial that (b1) implies (b2), and it follows by taking
$h:={{}^{*}\Theta_{A}}$ and using Lemma 2.13(f,d) that (b) is true. Thus (b),
(b1) and (b2) are equivalent.

If (c) is true then (c1) follows by taking $h:=\Phi_{A}$ and using Lemma
2.13(b,f). It is trivial that (c1) implies (c2). If $h\in{\cal PC}(B)$ and
$h\geq\Phi_{A}$ on $B$ then ${\Phi_{A}}^{*}\geq h^{*}$ on $B^{*}$, and (c2)
implies (c). Thus (c), (c1) and (c2) are equivalent. □

5 Applications of Section 4 to $E\times E^{*}$

In this section, we suppose that $E$ is a nonzero Banach space, and show how
the results of Section 4 can be applied to Example 4.4. We refer the reader to
Section 3 for the definitions of $\Theta_{A}$ and $\Phi_{A}$ in this case.

Remark 5.1. Before proceeding with our analysis, we make some remarks about
the essential difference between the concepts of MAS function introduced in
Definition 4.8 and VZ function introduced in Definition 2.5. As observed in
Example 4.4, we have $(E\times E^{*})^{*}=E^{**}\times E^{*}$ and
$\widetilde{q}(x^{**},x^{*})=\langle x^{*},x^{**}\rangle$, so we have all the
information needed to decide whether a function $f\in{\cal PC}(E\times E^{*})$
is an MAS function. The situation with VZ functions is different since that
involves the function $g$ in an essential way, and this is determined by the
precise norm we are using on $E\times E^{*}$. In order to clarify the
situation, we make the following definition.

Definition 5.2. We say that the norm $\|\cdot\|$ on $E\times E^{*}$ is special
if, for some $\tau>0$, $\|\cdot\|$ is identical with one of the norms
$\|\cdot\|_{1,\tau}$, $\|\cdot\|_{2,\tau}$ or $\|\cdot\|_{\infty,\tau}$
introduced in Example 2.4. As we pointed out in the comments in Example 4.4,
if $E\times E^{*}$ is normed by a special norm then $\iota(E\times E^{*})$ is
$\widetilde{p}$–dense in $(E\times E^{*})^{*}$.

Theorem 5.3. Let $E$ be a nonzero Banach space and $f\in{\cal PC}(E\times
E^{*})$ be a VZ function with respect to a given special norm on $E\times
E^{*}$. Then $f$ is a VZ function with respect to all special norms on
$E\times E^{*}$.

Proof. This is clear from the comments above and Theorem 4.9(c). □

Definition 5.4. Let Let $E$ be a nonzero Banach space and $f\in{\cal
PC}(E\times E^{*})$. We say that $f$ is a VZ function on $E\times E^{*}$ if
$f$ is a VZ function with respect to any one special norm on $E\times E^{*}$
or, equivalently, with respect to all special norms on $E\times E^{*}$. This
is also equivalent to the statement that $f$ is an MAS function.

Theorem 5.5(a) was obtained in [15, Theorem 8] under the VZ hypothesis and, in
[4, Theorem 4.2(2)] under the MAS hypothesis.

Theorem 5.5(c) extends the result proved in [15, Corollary 25] that ${\cal
M}f$ is of type (ANA).

Theorem 5.5(d) extends the result proved in [4, Theorem 4.2(2)].

Theorem 5.5(f) was obtained in [15, Corollary 7]. This is a very significant
result, because maximally monotone sets $A$ of $E\times E^{*}$ are known such
that $\overline{\pi_{E^{*}}(A)}$ is not convex. (The first such example was
given by Gossez in [3, Proposition, p. 360]). Thus (as was first observed in
[15]) Theorem 5.5(f) implies that there exist maximally monotone sets $A$ that
are not of the form ${\cal M}f$ for any lower semicontinuous VZ function on
$E\times E^{*}$ or, equivalently, not of the form ${\cal M}f$ for any lower
semicontinuous MAS function on $E\times E^{*}$. Theorem 5.5(f) can also be
proved directly from Lemma 2.7(a) rather than from the more circuitous
argument given here.

The techniques used in Theorem 5.5 originated in the negative alignment
analysis of [10, Section 8, pp. 274–280] and [13, Section 42, pp. 161–167].

Theorem 5.5. Let $E$ be a nonzero Banach space and $f\in{\cal PCLSC}(E\times
E^{*})$. Assume either that $f$ is a VZ function on $E\times E^{*}$ or,
equivalently, that $f$ is an MAS function. Then:

(a) ${\cal M}f$ is a maximally monotone subset of $E\times E^{*}$. (b) Let
$(x,x^{*})\in E\times E^{*}$ and $\alpha,\beta>0$. Then there exists a unique
value of $\omega\geq 0$ for which there exists a bounded sequence
$\big{\\{}(y_{n},y_{n}^{*})\big{\\}}_{n\geq 1}$ of elements of ${\cal M}f$
such that,

$\lim_{n\to\infty}\|y_{n}-x\|=\alpha\omega,\quad\lim_{n\to\infty}\|y_{n}^{*}-x^{*}\|=\beta\omega\quad\hbox{and}\quad\lim_{n\to\infty}\langle
y_{n}-x,y_{n}^{*}-x^{*}\rangle=-\alpha\beta\omega^{2}.$

(c) Let $(x,x^{*})\in E\times E^{*}\setminus{\cal M}f$ and $\alpha,\beta>0$.
Then there exists a bounded sequence
$\big{\\{}(y_{n},y_{n}^{*})\big{\\}}_{n\geq 1}$ of elements of ${\cal
M}f\cap\big{[}(E\setminus\\{x\\})\times(E^{*}\setminus\\{x^{*}\\})\big{]}$
such that,

$\lim_{n\to\infty}{\|y_{n}-x\|\over\|y_{n}^{*}-x^{*}\|}={\alpha\over\beta}\quad\hbox{and}\quad\lim_{n\to\infty}{\langle
y_{n}-x,y_{n}^{*}-x^{*}\rangle\over\|y_{n}-x\|\|y_{n}^{*}-x^{*}\|}=-1.$ $None$

In particular, ${\cal M}f$ is of type (ANA) (see [13, Definition 36.11, p.
152]). (d) Let $(x,x^{*})\in E\times E^{*}\setminus{\cal M}f$,
$\alpha,\beta>0$ and $\inf_{(y,y^{*})\in{\cal M}f}\langle
y-x,y^{*}-x^{*}\rangle>-\alpha\beta$. Then there exists a bounded sequence
$\big{\\{}(y_{n},y_{n}^{*})\big{\\}}_{n\geq 1}$ in ${\cal
M}f\cap\big{[}(E\setminus\\{x\\})\times(E^{*}\setminus\\{x^{*}\\})\big{]}$
such that (5.5.1) is satisfied, $\lim_{n\to\infty}\|y_{n}-x\|<\alpha$ and
$\lim_{n\to\infty}\|y_{n}^{*}-x^{*}\|<\beta$. In particular, ${\cal M}f$ is of
type (BR) (see [13, Definition 36.13, p. 153]). (e) Let $(x,x^{*})\in E\times
E^{*}\setminus{\cal M}f$, $\alpha,\beta>0$ and $f(x,x^{*})<\langle
x,x^{*}\rangle+\alpha\beta$. Then there exists a bounded sequence
$\big{\\{}(y_{n},y_{n}^{*})\big{\\}}_{n\geq 1}$ of elements of ${\cal
M}f\cap\big{[}(E\setminus\\{x\\})\times(E^{*}\setminus\\{x^{*}\\})\big{]}$
such that (5.5.1) is satisfied, $\lim_{n\to\infty}\|y_{n}-x\|<\alpha$ and
$\lim_{n\to\infty}\|y_{n}^{*}-x^{*}\|<\beta$. (f) We define the projection
maps $\pi_{E}\colon E\times E^{*}\to E$ and $\pi_{E^{*}}\colon E\times
E^{*}\to E^{*}$ by $\pi_{E}(x,x^{*}):=x$ and $\pi_{E^{*}}(x,x^{*}):=x^{*}$.
Then $\overline{\pi_{E}({\cal M}f)}=\overline{\pi_{E}(\hbox{\rm dom}\,f)}$ and
$\overline{\pi_{E^{*}}({\cal M}f)}=\overline{\pi_{E^{*}}(\hbox{\rm dom}\,f)}$.
Consequently, the sets $\overline{\pi_{E}({\cal M}f)}$ and
$\overline{\pi_{E^{*}}({\cal M}f)}$ are convex.

Proof. (a) is immediate from Theorem 2.9(c).

(b) Let $\tau:=\sqrt{\beta/\alpha}$ and use the norm $\|\cdot\|_{\infty,\tau}$
on $E\times E^{*}$. Lemma 2.7(b) provides us with a bounded sequence
$\big{\\{}(y_{n},y_{n}^{*})\big{\\}}_{n\geq 1}$ of elements of ${\cal M}f$
such that

$\lim_{n\to\infty}\big{[}\beta\|y_{n}-x\|^{2}/\alpha\vee\alpha\|y_{n}^{*}-x^{*}\|^{2}/\beta+\langle
y_{n}-x,y_{n}^{*}-x^{*}\rangle\big{]}=0.$

By passing to an appropriate subsequence, we can and will suppose that the
three limits $\rho:=\lim_{n\to\infty}\|y_{n}-x\|$,
$\sigma:=\lim_{n\to\infty}\|y_{n}^{*}-x^{*}\|$ and $\lim_{n\to\infty}\langle
y_{n}-x,y_{n}^{*}-x^{*}\rangle$ all exist. Consequently,
$\beta\rho^{2}/\alpha\vee\alpha\sigma^{2}/\beta+\lim_{n\to\infty}\langle
y_{n}-x,y_{n}^{*}-x^{*}\rangle=0$, from which

$\beta\rho^{2}/\alpha\vee\alpha\sigma^{2}/\beta=-\lim\nolimits_{n\to\infty}\langle
y_{n}-x,y_{n}^{*}-x^{*}\rangle\leq\rho\sigma=\sqrt{\beta\rho^{2}/\alpha}\sqrt{\alpha\sigma^{2}/\beta}.$

It follows easily from this that $\beta\rho^{2}/\alpha=\alpha\sigma^{2}/\beta$
and $\lim_{n\to\infty}\langle y_{n}-x,y_{n}^{*}-x^{*}\rangle=-\rho\sigma$. The
first of these equalities implies that $\rho/\alpha=\sigma/\beta$. We take
$\omega:=\rho/\alpha=\sigma/\beta$, and it is immediate that $\omega$ has the
required properties. The uniqueness of $\omega$ was established in [10,
Theorem 8.4(b), p. 276] and [13, Theorem 42.2(b), pp. 163–164].

(c) Following on from (b), if $\omega=0$ then $(\rho,\sigma)=(0,0)$, that is
to say $\lim_{n\to\infty}y_{n}=x$ in $E$ and
$\lim_{n\to\infty}y_{n}^{*}=x^{*}$ in $E^{*}$. Since ${\cal M}f$ is closed,
this would contradict the hypothesis that $(x,x^{*})\not\in{\cal M}f$. Thus
$\omega>0$, from which $\rho>0$ and $\sigma>0$. (c) now follows by truncating
the sequences so that, for all $n$, $\|y_{n}-x\|>0$ and
$\|y_{n}^{*}-x^{*}\|>0$.

(d) Continuing with the notation of (c), we have

$-\alpha\beta<\inf\nolimits_{(y,y^{*})\in{\cal M}f}\langle
y-x,y^{*}-x^{*}\rangle\leq\lim\nolimits_{n\to\infty}\langle
y_{n}-x,y_{n}^{*}-x^{*}\rangle=-\rho\sigma,$

from which $(\rho/\alpha)(\sigma/\beta)<1$. Since $\rho/\alpha=\sigma/\beta$,
in fact $\rho/\alpha<1$. and $\sigma/\beta<1$, that is to say
$\rho=\lim_{n\to\infty}\|y_{n}-x\|<\alpha$ and
$\sigma=\lim_{n\to\infty}\|y_{n}^{*}-x^{*}\|<\beta$. This gives (d). (e) is
immediate from (d) and the comment in Remark 2.17 that, for all $(x,x^{*})\in
E\times E^{*}$, $-\inf\nolimits_{(y,y^{*})\in{\cal M}f}\langle
y-x,y^{*}-x^{*}\rangle\leq f(x,x^{*})-\langle x,x^{*}\rangle$. . (f) If
$x\in\pi_{E}(\hbox{\rm dom}\,f)$ then there exists $x^{*}\in E^{*}$ such that
$f(x,x^{*})<\infty$, and so it follows from (e) that there exists
$(y,y^{*})\in{\cal M}f$ such that $\|y-x\|<1/n$. Consequently,
$x\in\overline{\pi_{E}({\cal M}f)}$. Thus we have proved that
$\pi_{E}(\hbox{\rm dom}\,f)\subset\overline{\pi_{E}({\cal M}f)}$. On the other
hand, ${\cal M}f\subset\hbox{\rm dom}\,f$, and so $\overline{\pi_{E}({\cal
M}f)}=\overline{\pi_{E}(\hbox{\rm dom}\,f)}$. We can prove in an exactly
similar way that $\overline{\pi_{E^{*}}({\cal
M}f)}=\overline{\pi_{E^{*}}(\hbox{\rm dom}\,f)}$. The convexity of the sets
$\overline{\pi_{E}({\cal M}f)}$ and $\overline{\pi_{E^{*}}({\cal M}f)}$ now
follows immediately. □

Remark 5.6. If we combine Theorem 2.9(a) (using the norm $\|\cdot\|_{2,1}$ on
$E\times E^{*}$) with the comments made in the proof of Theorem 5.5(e) we
obtain the following result: Let $E$ be a nonzero Banach space, $f\in{\cal
PCLSC}(E\times E^{*})$, and $f$ be a VZ function on $E\times E^{*}$. Then, for
all $(x,x^{*})\in E\times E^{*}$,

$\eqalign{\inf\nolimits_{(y,y^{*})\in{\cal
M}f}\sqrt{\|y-x\|^{2}+\|y^{*}-x^{*}\|^{2}}&\leq\sqrt{2}\sqrt{-\inf\nolimits_{(y,y^{*})\in{\cal
M}f}\langle y-x,y^{*}-x^{*}\rangle}\cr&\leq\sqrt{2}\sqrt{f(x,x^{*})-\langle
x,x^{*}\rangle}.}$

This strengthens the result proved in [15, Theorem 4], namely that

$\inf\nolimits_{(y,y^{*})\in{\cal
M}f}\sqrt{\|y-x\|^{2}+\|y^{*}-x^{*}\|^{2}}\leq 2\sqrt{f(x,x^{*})-\langle
x,x^{*}\rangle}.$

As we observed in Remark 2.17, the constant $\sqrt{2}$ is sharp.

Definition 5.7. Let $E$ be a nonzero Banach space and $A$ be a nonempty
monotone subset of $E\times E^{*}$. We say that $A$ is of type (NI) if, for
all $(x^{**},x^{*})\in E^{**}\times E^{*}$,

$\inf\nolimits_{(s,s^{*})\in A}\langle
x^{*}-s^{*},x^{**}-\widehat{s}\rangle\leq 0.$

This concept was introduced in [9, Definition 10, p. 183]. We say that $A$ is
strongly representable if there exists $f\in{\cal PCLSC}(E\times E^{*})$ such
that $f\geq q$ on $E\times E^{*}$, $f^{*}\geq\widetilde{q}$ on $E^{**}\times
E^{*}$ (i.e., $f$ is a lower semicontinuous MAS function) and ${\cal M}f=A$.
This concept was introduced and studied in [4], [5] and [15].

Theorem 5.8 was motivated by and extends that proved in [5, Theorem 1.2]. The
most significant part of it is the fact that (a) implies (d) and (a) implies
(e). In particular, if $A$ is maximally monotone of type (NI), then the
conclusions of Theorem 5.5(b–f) hold (with ${\cal M}f$ replaced by $A$). This
leads to a substantial generalization of [10, Theorem 8.6, pp. 277–278] and
[13, Theorem 42.6, pp. 163–164]. The fact that $\overline{\pi_{E}A}$ and
$\overline{\pi_{E^{*}}A}$ are convex whenever $A$ is of type (NI) was first
proved by Zagrodny in [14].

Theorem 5.8. Let $E$ be a nonzero Banach space and $A$ be a maximally monotone
subset of $E\times E^{*}$. Then the following conditions are equivalent: (a)
$A$ is of type (NI). (b) For all $(x^{**},x^{*})\in E^{**}\times E^{*}$,
$\sup\nolimits_{(s,s^{*})\in A}\big{[}\langle s,x^{*}\rangle+\langle
s^{*},x^{**}\rangle-\langle s,s^{*}\rangle\big{]}\geq\langle
x^{*},x^{**}\rangle$. (c) For all $(x^{**},x^{*})\in E^{**}\times E^{*}$,
$\sup\nolimits_{(y,y^{*})\in E\times E^{*}}\big{[}\langle
y,x^{*}\rangle+\langle y^{*},x^{**}\rangle-\Phi_{A}(y,y^{*})\big{]}\geq\langle
x^{*},x^{**}\rangle$. (d) $A$ is strongly representable. (e) There exists a
lower semicontinuous VZ function on $E\times E^{*}$ such that ${\cal M}f=A$.
(f) $\Phi_{A}$ is a VZ function on $E\times E^{*}$. (g) ${}^{*}\Theta_{A}$ is
a VZ function on $E\times E^{*}$. (b1) If $h\in{\cal PC}(E\times E^{*})$ and
${}^{*}\Theta_{A}\geq h$ on $E\times E^{*}$ then, for all $(x^{**},x^{*})\in
E^{**}\times E^{*}$,

$h^{*}(x^{**},x^{*})=\sup\nolimits_{(y,y^{*})\in E\times E^{*}}\big{[}\langle
y,x^{*}\rangle+\langle y^{*},x^{**}\rangle-h(y,y^{*})\big{]}\geq\langle
x^{*},x^{**}\rangle.$ $None$

(b2) If $h\in{\cal PCLSC}(E\times E^{*})$ and ${}^{*}\Theta_{A}\geq
h\geq\Phi_{A}$ on $E\times E^{*}$ then, for all $(x^{**},x^{*})\in
E^{**}\times E^{*}$, (5.8.1) is satisfied. (c1) There exists $h\in{\cal
PCLSC}(E\times E^{*})$ such that ${}^{*}\Theta_{A}\geq h\geq\Phi_{A}$ on
$E\times E^{*}$ and, for all $(x^{**},x^{*})\in E^{**}\times E^{*}$, (5.8.1)
is satisfied. (c2) There exists $h\in{\cal PC}(E\times E^{*})$ such that
$h\geq\Phi_{A}$ on $E\times E^{*}$ and, for all $(x^{**},x^{*})\in
E^{**}\times E^{*}$, (5.8.1) is satisfied.

Proof. These results are all immediate from the corresponding parts of Theorem
4.10. □

6 Appendix: a nonhausdorff Fenchel–Moreau theorem

In Remark 2.14, we referred to the Fenchel–Moreau theorem for (possibly
nonhausdorff) locally convex spaces. We shall give a proof of this result in
Theorem 6.1. When we say that $X$ is a locally convex space, we mean that $X$
is a nonzero real vector space endowed with a topology compatible with its
vector structure with a base of neighborhoods of $0$ of the form
$\big{\\{}x\in X\colon\ S(x)\leq 1\big{\\}}_{S\in{\cal S}(X)}$, where ${\cal
S}(X)$ is a family of seminorms on $X$ such that if $S_{1}\in{\cal S}(X)$ and
$S_{2}\in{\cal S}(X)$ then $S_{1}\vee S_{2}\in{\cal S}(X)$; and if $S\in{\cal
S}(X)$ and $\lambda\geq 0$ then $\lambda S\in{\cal S}(X)$. If $L$ is a linear
functional on $X$ then $L$ is continuous if, and only if, there exists
$S\in{\cal S}(X)$ such that $L\leq S$ on $X$.

As an example of the construction above, we can suppose that $X$ and $Y$ are
vector spaces paired by a bilinear form $\langle\cdot,\cdot\rangle$. Then
$\big{(}X,w(X,Y)\big{)}$ is a locally convex space with determining family of
seminorms
$\big{\\{}|\langle\cdot,y_{1}\rangle|\vee\cdots\vee|\langle\cdot,y_{n}\rangle|\big{\\}}_{n\geq
1,\ y_{1},\dots,y_{n}\in Y}$.

The author is grateful to Constantin Zălinescu for showing him a proof of
Theorem 6.1 based on the standard (Hausdorff) result and a quotient
construction. The proof we give here is a simplification of the result on
“Fenchel–Moreau points” of [12, Theorem 5.3, pp. 157–158] or [13, Theorem
12.2, pp. 59–60], which is also valid in the nonhausdorff setting.

Theorem 6.1. Let $X$ be a locally convex space and $f\in{\cal PC}(X)$ be lower
semicontinuous. Write $X^{*}$ for the set of continuous linear functionals on
$X$. If $L\in X^{*}$, define $f^{*}(L):=\sup_{X}\big{[}L-f\big{]}$. Let $y\in
X$. Then

$f(y)=\sup\nolimits_{L\in X^{*}}\big{[}L(y)-f^{*}(L)\big{]}.$ $None$

Proof. Since, for all $L\in X^{*}$, $L(y)-f^{*}(L)=\inf\nolimits_{x\in
X}\big{[}L(y)-L(x)+f(x)\big{]}=(f\mathop{\nabla}L)(y)$ and the inequality
“$\geq$” in (6.1.1) is obvious from the definition of $f^{*}(L)$, we only have
to prove that

$f(y)\leq\sup\nolimits_{L\in X^{*}}(f\mathop{\nabla}L)(y)\big{]}.$ $None$

Let $\lambda\in\hbox{\tenmsb R}$ and $\lambda<f(y)$. Since $f$ is proper,
there exists $z\in\hbox{\rm dom}\,f$. Choose $Q\in{\cal S}(X)$ such that

$Q(z-x)\leq 1\quad\Longrightarrow\quad f(x)>f(z)-1$ $None$

and

$Q(y-x)\leq 1\quad\Longrightarrow\quad f(x)>\lambda.$ $None$

We first prove that

$(f\mathop{\nabla}Q)(z)\geq f(z)-1.$ $None$

To this end, let $x$ be an arbitrary element of $X$. If $Q(z-x)\leq 1$ then
(6.1.3) implies that $f(x)+Q(z-x)\geq f(x)>f(z)-1$. If, on the other hand,
$Q(z-x)>1$, let $\gamma:=1/Q(z-x)\in\,]0,1[\,$ and put $u:=\gamma
x+(1-\gamma)z$. Then $Q(z-u)=\gamma Q(z-x)=1$ and so, from the convexity of
$f$, and (6.1.3) with $x$ replaced by $u$,

$\gamma f(x)+(1-\gamma)f(z)\geq f\big{(}\gamma
x+(1-\gamma)z\big{)}=f(u)>f(z)-1.$

Substituting in the formula for $\gamma$ and clearing of fractions yields
$f(x)+Q(z-x)\geq f(z)$. This completes the proof of (6.1.5).

Now let $M\geq 1$ and $M\geq\lambda+2+Q(z-y)-f(z)$. We will prove that

$(f\mathop{\nabla}MQ)(y)\geq\lambda.$ $None$

To this end, let $x$ be an arbitrary element of $X$. If $Q(y-x)\leq 1$ then
(6.1.4) implies that $f(x)+MQ(y-x)\geq f(x)>\lambda$. If, on the other hand,
$Q(y-x)>1$ then, from (6.1.5),

$\eqalign{f(x)+MQ(y-x)&=f(x)+Q(y-x)+(M-1)Q(y-x)\cr&\geq
f(x)+Q(z-x)-Q(z-y)+(M-1)\cr&\geq f(z)-1-Q(z-y)+M-1\geq\lambda,}$

which completes the proof of (6.1.6). The “Hahn–Banach–Lagrange theorem”
of[12, Theorem 2.9, p. 153] or [13, Theorem 1.11, p. 21] now provides us with
a linear functional $L$ on $X$ such that $L\leq MQ$ on $X$ and
$(f\mathop{\nabla}L)(y)\geq\lambda$. (6.1.2) now follows by letting
$\lambda\to f(y)$. □

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Department of Mathematics

University of California

Santa Barbara

CA 93106-3080

U. S. A.

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