Learning Materials
Features
Discover
Chapter 12: Problem 21
Show that if \(C(K)\) is reflexive, then \(K\) is finite.
Over 30 million students worldwide already upgrade their learning with 91影视!
These are the key concepts you need to understand to accurately answer the question.
All the tools & learning materials you need for study success - in one app.
Get started for free
Let \(X\) be a separable Banach space and assume that the dual norm of \(X^{*}\) is G芒teaux differentiable. Show that every element of \(X^{* *}\) is a first Baire class function when considered as a function on \(\left(B_{X^{*}}, w^{*}\right)\).
Let \(K\) be a scattered compact. Show that every weakly compact operator from \(C(K)\) into a Banach space \(X\) is a compact operator.
Let \(K=\Gamma \cup\\{\infty\\}\) be an Alexandrov compactification of the discrete space \(\Gamma\). Show that \(K\) is a scattered Eberlein compact and \(C(K)\) is isomorphic to \(c_{0}(\Gamma)\). Note that \(K\) is a uniform Eberlein compact by Theorem \(12.17 .\)
Find an example of a compact set \(K\) such that \(C(K)\) is nonseparable but does not contain \(c_{0}(\Gamma)\) for any \(\Gamma\) uncountable.
Show that there is an equivalent norm on \(\ell_{\infty}\) such that its dual ball is not \(w^{*}\) -separable, although the standard unit ball of \(\ell_{\infty}=c_{0}^{* *}\) is \(w^{*}\) -separable by Goldstine's theorem. Note that there is a Banach space \(X\) such that its dual is \(w^{*}\) -separable and the dual ball of no equivalent norm on \(X\) is \(w^{*}\) -separable ([JoL1]). Note also that the dual space \(C(K)^{*}\), where \(K\) is Kunen's compact, has the property that every subset of \(C(K)^{*}\) is \(w^{*}\) -separable.
What do you think about this solution?
We value your feedback to improve our textbook solutions.
Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine