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Locally Convex Topological Vector Spaces
Published in Kenneth Kuttler, Modern Analysis, 2017
4. ↑ We can define a topological vector space to be locally convex if it has a local basis at 0 of convex sets. Show that if X is locally convex, then it has a local basis at 0 of convex, balanced open sets. Hint: If 0 ∈ U, and U is convex, Problem 1 shows there exists an open balanced set W such that 0 ∈ W ⊆ U. Thus
Efficient solutions in generalized linear vector optimization
Published in Applicable Analysis, 2019
Let X be a locally convex Hausdorff topological vector space with the dual space denoted by . For any and , indicates the value of at x. For a subset of a locally convex Hausdorff topological vector space, we denote its interior by , and its topological closure by .
Prox-regular sets and Legendre-Fenchel transform related to separation properties
Published in Optimization, 2022
Samir Adly, Florent Nacry, Lionel Thibault
It has been well-recognized that the geometric Hahn-Banach theorems are among of the most important and powerful principles of functional analysis (see, e.g. [5,6,29,30] and the references therein). Roughly speaking, the geometric Hahn-Banach theorem for closed convex sets asserts that a compact convex set A and a closed convex set B of X (or more generally of a locally convex space) with can be separated by a hyperplane/half-space. The case where A is reduced to a singleton (so A is an exterior point of B) is of a great interest and can be stated as follows (see also [29, Theorem 6.23]).
L 0-convex compactness and its applications to random convex optimization and random variational inequalities
Published in Optimization, 2021
Tiexin Guo, Erxin Zhang, Yachao Wang, Mingzhi Wu
It is well known that a locally convex space can be defined in two equivalent ways – one by a family of seminorms and the other by a base of convex neighbourhoods. Guo [19] gave a random generalization of the first kind by means of a family of -seminorms. In 2009, motivated by financial applications Filipović, Kupper and Vogelpoth [23] gave a random generalization of the second kind by means of a family of -convex neighbourhoods, which leads to the notion of a locally -convex module as well as another kind of topology for a random locally convex module, called the locally -convex topology. The central purpose of [23] is an attempt to establish random convex analysis, providing an analytical basis for conditional convex risk measures, but such an attempt is realized by Guo et al. in [24–26]. Following [23], Guo [27] introduced the notion of the countable concatenation property (also called σ-stability or stability) for a subset of an -module in order to establish the inherent connections between the two theories derived from the two kinds of topologies (namely the -topology and the locally -convex topology) for a random locally convex space. Based on [27], a basic random convex analysis was established in [24–26] with applications to conditional convex risk measures [28]. Guo's work [27] also stimulated a series of subsequent researches [29–39], in particular, the notion of the countable concatenation property was frequently employed in [29,31,34] so that a great number of basic results in real analysis and linear algebra can be generalized from Euclidean spaces to random Euclidean spaces.