91Ó°ÊÓ

Let \(x, x_{n} \in \ell_{2}\) be such that \(x_{n} \stackrel{w}{\rightarrow} x\) in \(\ell_{2}\). Show that there is a subsequence \(\left\\{x_{n_{k}}\right\\}\) such that the Cesàro means $$ \frac{x_{n_{1}}+x_{n_{2}}+\ldots+x_{n_{k}}}{k} $$ converge to \(x\) in \(\ell_{2}\) (the Banach-Saks theorem). Show that an analogous statement is true for \(c_{0}\). Spaces that have this property are called spaces with the weak BanachSaks property.

Prove directly that if \(X\) is a separable Banach space, then \(X^{*}\) is \(w^{*}\) -separable.

Let \(X\) be an infinite-dimensional Banach space. Show that the \(w\) topology of \(X\) is not first countable; in particular, it is not metrizable. The same is true for the \(w^{*}\) -topology of \(X^{*}\).

Let \(X\) be a reflexive Banach space. Show that if \(Y\) is isomorphic to \(X\), then \(Y\) is reflexive.

Let \(Y\) be a closed subspace of a reflexive Banach space \(X .\) Show that \(X / Y\) is reflexive.

Show that none of } C[0,1], c_{0}, L_{1}[0,1] \text { are isometric to a dual space }

Prove the following Smulian's theorem: A Banach space \(X\) is reflexive if each nested sequence \(C_{n} \supset C_{n+1}\) of closed convex subsets of \(B_{X}\) has a nonempty intersection.

Let \(X\) be a Banach space, and let \(C\) be a separable set in \(X^{*} .\) If \(C\) is \(w^{*}\) -compact, is \(D=\overline{\operatorname{conv}}(C)\) also \(w^{*}\) -compact? What if the separability assumption is dropped?