Abstract
We first prove Mazur’s lemma in a random locally convex module endowed with the locally L0-convex topology. Then, we establish the embedding theorem of an L0-prebarreled random locally convex module, which says that if (S,P) is an L0-prebarreled random locally convex module such that S has the countable concatenation property, then the canonical embedding mapping J of S onto J(S)⊂(Ss⁎)s⁎ is an L0-linear homeomorphism, where (Ss⁎)s⁎ is the strong random biconjugate space of S under the locally L0-convex topology.
Highlights
Mazur’s lemma in a locally convex space is a very useful fact in convex analysis
Based on the idea of randomizing functional space theory, a new approach to random functional analysis was initiated by Guo in [1,2,3]; in particular, the study of random normed modules and random inner product modules together with their random conjugate spaces was already the central theme in random functional analysis in [2, 3]
Random normed modules, random inner product modules, random locally convex modules, and the theory of random conjugate spaces still occupy a central place in random functional analysis
Summary
Mazur’s lemma in a locally convex space is a very useful fact in convex analysis. The embedding theorem of a locally convex space into its biconjugate space has played a crucial role in the study of semireflexivity and reflexivity of a locally convex space. In 2009, Guo et al first proved Mazur’s lemma in a random locally convex module endowed with the (ε, λ)-
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