[Paper Review] Weak compactness of sublevel sets in complete locally convex spaces

Pedro Pérez-Aros, Lionel Thibaul|arXiv (Cornell University)|Jan 23, 2018

Advanced Topology and Set Theory15 references3 citations

TL;DR

This paper establishes a weak compactness criterion for sublevel sets of convex functions in complete locally convex spaces by showing that if $ f - x^* $ attains its minimum for all $ x^* $ in an open Mackey-topology set $ U eq abla $, then the sets $ \{x \in X : f(x) - \langle x^*, x \rangle \leq \gamma\} $ are relatively weakly compact for all $ \gamma \in \mathbb{R} $ and $ x^* \in U $. The result extends James-type compactness theorems beyond Banach spaces and links epi-pointed functions to inf-compactness via conjugate analysis.

ABSTRACT

In this work we prove that if $X$ is a complete locally convex space and $f:X o \mathbb{R}\cup \{+\infty \}$ is a function such that $f-x^\ast$ attains its minimum for every $x^\ast \in U$, where $U$ is an open set with respect to the Mackey topology in $X^\ast$, then for every $γ\in \mathbb{R}$ and $x^\ast \in U$ the set $\{ x\in X : f(x)- \langle x^\ast , x angle \leq γ\}$ is relatively weakly compact. This result corresponds to an extension of Theorem 2.4 in [J. Saint Raymond, Mediterr. J. Math. 10 (2013), no. 2, 927--940]. Directional James compactness theorems are also derived.

Motivation & Objective

• To extend Theorem 1.2 (James-type weak compactness in Banach spaces) to complete locally convex spaces.
• To characterize weak compactness of sublevel sets $ \{x \in X : f(x) - \langle x^*, x \rangle \leq \gamma\} $ in terms of the existence of minimizers for $ f - x^* $ on an open Mackey-topology set $ U \subseteq X^* $.
• To clarify the connection between epi-pointed functions and the variational property that $ f - x^* $ attains its minimum for all $ x^* \in U $, where $ U $ is open in the Mackey topology.
• To derive directional James compactness theorems for sets using convex analysis and inf-convolution techniques.

Proposed method

• Use of the Moreau-Rockafellar subdifferential and its inverse to relate minimizers of $ f - x^* $ to elements in $ \partial f^*(x^*) $.
• Application of the Mackey topology $ \tau(X^*, X) $ to define open sets $ U \subseteq X^* $ where $ f - x^* $ is assumed to attain its minimum.
• Employment of inf-convolution and conjugate functions to analyze the structure of sublevel sets and their weak compactness.
• Leveraging the fact that $ f^* $ is finite and continuous on $ U $ in the Mackey topology to deduce weak compactness of sublevel sets.
• Use of directional asymptotic cones and polar sets to characterize weak compactness in terms of the geometry of the dual space.
• Proof technique based on Moors’ shorter argument for Theorem 1.2, adapted to the locally convex setting via convex duality and topological tools.

Experimental results

Research questions

• RQ1 Under what conditions on a proper lower semicontinuous convex function $ f $ on a complete locally convex space $ X $ are the sets $ \{x \in X : f(x) - \langle x^*, x \rangle \leq \gamma\} $ relatively weakly compact for all $ \gamma \in \mathbb{R} $ and $ x^* \in U $, where $ U $ is open in the Mackey topology on $ X^* $?
• RQ2 How does the assumption that $ f - x^* $ attains its minimum for all $ x^* \in U $, with $ U $ open in the Mackey topology, relate to the weak compactness of sublevel sets?
• RQ3 What is the precise relationship between epi-pointed functions and the variational property of minimum attainment for $ f - x^* $ on an open Mackey set?
• RQ4 Can directional James compactness theorems be derived for sets in complete locally convex spaces using convex analysis?
• RQ5 What is the role of the Mackey topology in ensuring weak compactness of sublevel sets, and how does it generalize the Banach space case?

Key findings

• If $ f - x^* $ attains its minimum for every $ x^* \in U $, where $ U \subseteq X^* $ is open in the Mackey topology, then the sublevel sets $ \{x \in X : f(x) - \langle x^*, x \rangle \leq \gamma\} $ are relatively weakly compact for all $ \gamma \in \mathbb{R} $ and $ x^* \in U $.
• The result extends Theorem 1.2 (Moors’ version of James’ theorem) from Banach spaces to complete locally convex spaces by replacing norm-based assumptions with Mackey-topological continuity of the conjugate function.
• The class of epi-pointed functions coincides with the class of functions for which $ f^* $ is Mackey-continuous on an open set $ U \subseteq X^* $, and this condition is equivalent to $ f - x^* $ being weakly inf-compact for all $ x^* \in U $.
• The sublevel sets $ \textnormal{lev}_{f - x^*}^{\leq}(\gamma) $ are relatively weakly compact even when $ f $ is not bounded below or when $ \textnormal{dom} f^* \neq X^* $, provided the minimum-attainment condition holds on an open Mackey set.
• The example of the norm function on a non-reflexive Banach space shows that the assumption $ \mathcal{E}(B^\circ) \neq \emptyset $ is essential, as $ f - x^* $ may attain its minimum everywhere but sublevel sets fail to be weakly compact.
• A complete locally convex space $ X $ is semi-reflexive if and only if $ \mathcal{E}(B^\circ) \neq \emptyset $ for every bounded, circled, convex set $ B \subseteq X $, linking weak compactness of bounded sets to the non-emptiness of the set of directions where $ f - x^* $ attains its minimum.