\def\bra#1#2{\left\langle#1,#2\right\rangle}
\def\co{\hbox{co}}
\def\dom{\hbox{dom}}
\def\ex{\hbox{there exists}}
\def\intr{\hbox{int}}
\def\st{\hbox{such that}}
\def\all{\hbox{for all}}
\def\r{\hbox{I \hskip - .5em R}}
\def\rbar{\r\cup\{\infty\}}
\magnification 1200
\centerline{\bf Convergence, Minimax and Monotonicity}
\bigskip
\centerline{S. Simons}
\bigskip
\centerline{\bf Abstract}
\medskip
\noindent Let $x_n$ be a bounded sequence of continuous functions on
[0,1] and,
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$$\all\ t \in [0,1],\quad \limsup_{n \to \infty}x_n(t) \le 0.$$
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Then, from Fatou's Lemma,
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$$\limsup_{n \to \infty}\int x_n \le 0.$$
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In the first talk, we give a proof of this result entirely without
measure theory, using only that the uniform limit of continuous
functions is continuous, and that every continuous function on [0,1]
attains its maximum.   The method of proof can be applied in situations
in functional analysis where measure theory is not available, and leads
to a generalization of Rainwater's theorem on weak convergence in a
Banach space.   Curiously, the proof depends on an eigenvector property
of a certain family of matrices.   The link between this talk and the
following one is that they are both closely related to the ``sup
theorem'' of R. C. James.
\medskip
\noindent Now let $E$ be a Banach space with dual $E^*$.   In the
second talk we first use the Hahn-Banach theorem in the version due to
Mazur and Orlicz to prove a simple minimax theorem.   It follows from
this that if $B$ is a nonempty convex $w(E,E^*)$-compact convex subset
of $E$ and $A$ is a nonempty bounded convex subset of $E^*$ then
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$$\max_{x \in B}\inf_{x^* \in A}\bra{x}{x^*} =
\inf_{x^* \in A}\max_{x \in B}\bra{x}{x^*}.$$
%
However, one cannot generalize the minimax theorem too much, since we
will also prove that if $B$ is a nonempty bounded closed convex subset
of $E$ and, for all nonempty bounded norm-closed convex subsets $A$ of
$E^*$,
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$$\sup_{x \in B}\inf_{x^* \in A}\bra{x}{x^*} \ge
\inf_{x^* \in A}\sup_{x \in B}\bra{x}{x^*}$$
%
then $B$ is $w(E,E^*)$--compact.
%
\medskip
\noindent In the remaining three talks, we discuss various
applications of the minimax theorem to monotone operators.
\medskip
\noindent In the third talk, we give some criteria for a monotone subset
of $E \times E^*$ to be maximal monotone when $E$ is reflexive, and
deduce from this a simple proof that if $E$ is reflexive, $S : E \to
2^{E^*}$ is maximal monotone and $J$ is the duality map then $S + J$ is
surjective.
\medskip
\noindent In the fourth talk, we shall show how $S$
determines a convex function $\chi_S : E \to \rbar$.   We shall prove
that if $S$ is maximal monotone then
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$$\intr D(S) = \intr\,(\dom\,\chi_S),$$
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and also that if $x \in E$ and,
%
$$\all\ w \in E \setminus \{0\},\quad \ex\ \delta > 0\ \st\quad x +
\delta w \in \co\,D(S)$$
%
then $x \in \intr D(S)$.   We will also discuss the connection between
the function $\chi_S$ and the local boundedness of monotone operators.
\medskip
\noindent In the fifth talk, we suppose that $S_1,\ S_2  : E \to
2^{E^*}$ are maximal monotone, and obtain necessary and sufficient
conditions for $S_1 + S_2$ to be maximal monotone when $E$ is
reflexive.   We deduce that $S_1 + S_2$ is maximal monotone when $E$ is
reflexive and
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$$\dom\chi_{S_1} - \dom\chi_{S_2}\quad \hbox{is absorbing.}$$
%
We shall also show the equivalence of various conditions
that have been given recently that imply the maximal monotonicity of
$S_1 + S_2$.
\medskip
\noindent We do not use any renorming theorems or any fixed-point
related results.
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\bigskip
\baselineskip 12pt
\parindent0pt
Department of Mathematics\par
University of California\par
Santa Barbara\par
CA 93106-3080\par
\bye


