Krein-Milman's Extreme Point Theorem and Weak Topology on
            Hyperspace

Hu, Jennifer Shueh-Inn; Hu, Thakyin

doi:10.11650/tjm.20.2016.6411

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2021

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“…This difficulty is overcome when we come up with a simple lemma that solves our problems and enables us to prove that Markov-Kakutani Theorem can be extended to the hyperspace (CC(X), h), which is the main result of this paper. Examples have been given showing that some results that are valid on the underlying Banach space X cannot be extended to the hyperspace (CC(X), h) [3,5]. Our theorem might provide a new tool to help solve the above mentioned problems.…”

Section: Introductionmentioning

confidence: 98%

Markov-Kakutani Theorem on Hyperspace of a Banach Space

Hu²

2021

Tamkang J. Math.

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Suppose $X$ is a Banach space and $K$ is a compact convex subset of $X$. Let $\mathcal{F}$ be a commutative family of continuous affine mappings of $K$ into $K$. It follows from Markov-Kakutani Theorem that $\mathcal{F}$ has a common fixed point in $K$. Suppose now $(CC(X), h)$ is the corresponding hyperspace of $X$ containing all compact, convex subsets of $X$ endowed with Hausdorff metric $h$. We shall prove the above version of Markov-Kakutani Theorem is valid on the hyperspace $(CC(X), h)$.

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Section: Introductionmentioning

confidence: 98%