Chapter 13

Prove that any Cauchy sequence that has a convergent subsequence must itself converge.

Let \(V\) be an inner product space and let \(A\) and \(B\) be subsets of \(V\). Show that a) \(A \subseteq B \Rightarrow B^{\perp} \subseteq A^{\perp}\) b) \(A^{\perp}\) is a closed subspace of \(V\) c) \([\operatorname{cspan}(A)]^{\perp}=A^{\perp}\)

Let \(V\) be an inner product space and \(S \subseteq V\). Under what conditions 15 \(S^{\perp \perp \perp}=S^{\perp}\) ?

Prove that a subspace \(S\) of a Hilbert space \(H\) is closed if and only if \(S=S^{\perp \perp}\).

Let \(V\) be the subspace of \(\ell^{2}\) consisting of all sequences of real numbers with the property that each sequence has only a finite number of nonzero terms. Thus, \(V\) is an inner product space. Let \(K\) be the subspace of \(V\) consisting of all sequences \(x=\left(x_{n}\right)\) in \(V\) with the property that \(\Sigma x_{n} / n=0\). Show that \(K\) is closed, but that \(K^{\perp 1} \neq K\). Hint: For the latter, show that \(K^{\perp}=\\{0\\}\) by considering the sequences \(u=(1, \ldots,-n, \ldots)\), where the term \(-n\) is in the \(n\)th coordinate position.

Let \(\mathcal{O}=\left\\{u_{1}, u_{2}, \ldots\right\\}\) be an orthonormal set in \(H\). If \(x=\Sigma r_{k} u_{k}\) converges, show that $$ \|x\|^{2}=\sum_{k=1}^{\infty}\left|r_{k}\right|^{2} $$

Prove that if an infinite series $$ \sum_{k=1}^{\infty} x_{k} $$ converges absolutely in a Hilbert space \(H\), then it also converges in the sense of the "net" definition given in this section.

Let \(\left\\{r_{k} \mid k \in K\right\\}\) be a collection of nonnegative real numbers. If the sum on the left below converges, show that $$ \sum_{k \in K} r_{k}=\sup _{J \operatorname{fin}_{j \in K}} \sum_{k \in J} r_{k} $$

Prove that if a Hilbert space \(H\) has infinite Hilbert dimension, then no Hilbert basis for \(H\) is a Hamel basis.

Prove that \(\ell^{2}(K)\) is a Hilbert space for any nonempty set \(K\).

Prove that any linear transformation between finite-dimensional Hilbert spaces is bounded.

Prove that if \(f \in H^{*}\), then \(\operatorname{ker}(f)\) is a closed subspace of \(H\).

Can a Hilbert space have countably infinite Hamel dimension?

Let \(\tau\) and \(\sigma\) be bounded linear operators on \(H\). Verify the following: a) \(\|r \tau\|=|r|\|T\|\) b) \(\|\tau+\sigma\| \leq\|\tau\|+\|\sigma\|\) c) \(\|\tau \sigma\| \leq\|\tau\|\|\sigma\|\)

Use the Riesz representation theorem to show that \(H^{*} \approx H\) for any Hilbert space \(H\).