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Magazine Problem 2
A Banach space \(X\) is reflexive if and only if \(X^{*}\) is reflexive.
Problem 6
Prove that the unit ball in a Banach space \(X\) is compact if and only if \(X\) is finite-dimensional.
Problem 7
If \(X\) is a separable Hilbert space, show that every bounded sequence contains a weakly convergent subsequence.
Problem 8
If \(X\) is a Hilbert space and \(S\) is any subset of \(X\), define $$ S^{\perp}=\\{y \in X:\langle x, y\rangle=0 \text { for all } x \in S\\} . $$ (a) Show that \(S^{\perp}\) is a closed subspace of \(X\). (b) Show that \(S \cap S^{\perp}\) can contain only the zero vector. (c) If \(S \subset T\) are both subsets of \(X\), show that \(T^{\perp} \subset S^{\perp} .\) (d) If \(\bar{S}\) is the closure of \(S\) in \(X\), show that \(S^{\perp}=\bar{S}^{\perp}\).
Problem 9
If \(T: X \rightarrow X\) is a self-adjoint compact linear operator on a Hilbert space \(X\), then either \(\|T\|\) or \(-\|T\|\) is an eigenvalue for \(T\).
