Elementary Hilbert Space Theory

If $M$ is a closed subspace of $H$, prove that $M = \left(M^{\perp }\right)^{\perp }$. Is there a similar true statement for subspaces $M$ which are not necessarily closed?

Let $\left\{ x_{n}: n = 1,2,3, \ldots \right\}$ be a linearly independent set of vectors in $H$. Show that the following construction yields an orthonormal set $\left\{ u_{n}\right\}$ such that $\left\{ x_{1}, \ldots , x_{N}\right\}$ and $\left\{ u_{1}, \ldots , u_{N}\right\}$ have the same span for all $N$.

Note that this leads to a proof of the existence of a maximal orthonormal set in separable Hilbert spaces which makes no appeal to the Hausdorff maximality principle. (A space is separable if it contains a countable dense subset).

Show that $L^{p}(T)$ is separable if $1 \leq p < \infty$, but that $L^{\infty }(T)$ is not separable.

Show that $H$ is separable if and only if $H$ contains a maximal orthonormal system which is at most countable.

If $M = \left\{ x: L x = 0\right\}$, where $L$ is a continuous linear functional on $H$, prove that $M^{\perp }$ is a vector space of dimension 1 (unless $M = H$).

Let $\left\{ u_{n}\right\} (n = 1,2,3, \ldots )$ be an orthonormal set in $H$. Show that this gives an example of a closed and bounded set which is not compact. Let $Q$ be the set of all $x \in H$ of the form

Prove that $Q$ is compact. ($Q$ is called the Hilbert cube).

More generally, let $\left\{ \delta_{n}\right\}$ be a sequence of positive numbers, and let $S$ be the set of all $x \in H$ of the form

Prove that $S$ is compact if and only if $\sum_{1}^{\infty } \delta_{n}^{2} < \infty$. Prove that $H$ is not locally compact.

Suppose $\left\{ a_{n}\right\}$ is a sequence of positive numbers such that $\sum a_{n} b_{n} < \infty$ whenever $b_{n} \geq 0$ and $\sum b_{n}^{2} < \infty$. Prove that $\sum a_{n}^{2} < \infty$.

Suggestion: If $\sum a_{n}^{2} = \infty$ then there are disjoint sets $E_{k}(k = 1,2,3, \ldots )$ so that

Define $b_{n}$ so that $b_{n} = c_{k} a_{n}$ for $n \in E_{k}$. For suitably chosen $c_{k}, \sum a_{n} b_{n} = \infty$ although $\sum b_{n}^{2} < \infty$.

If $H_{1}$ and $H_{2}$ are two Hilbert spaces, prove that one of them is isomorphic to a subspace of the other. (Note that every closed subspace of a Hilbert space is a Hilbert space).

If $A \subset [0,2 \pi ]$ and $A$ is measurable, prove that

Let $n_{1} < n_{2} < n_{3} < \cdots$ be positive integers, and let $E$ be the set of all $x \in [0,2 \pi ]$ at which $\left\{ \sin n_{k} x\right\}$ converges. Prove that $m(E) = 0$. Hint: $2 \sin^{2} \alpha = 1-\cos 2 \alpha$, so $\sin n_{k} x \rightarrow \pm 1 / \sqrt{2}$ a.e. on $E$, by Exercise 9.

Find a nonempty closed set $E$ in $L^{2}(T)$ that contains no element of smallest norm.

The constants $c_{k}$ in Sec. 4.24 were shown to be such that $k^{-1} c_{k}$ is bounded. Estimate the relevant integral more precisely and show that

Suppose $f$ is a continuous function on $\mathbb {R}^{1}$, with period 1. Prove that

for every irrational real number $\alpha$. Hint : Do it first for

Compute

and find

where $g$ is subject to the restrictions

Compute

State and solve the corresponding maximum problem, as in Exercise 14.

If $x_{0} \in H$ and $M$ is a closed linear subspace of $H$, prove that

Show that there is a continuous one-to-one mapping $\gamma$ of $[0,1]$ into $H$ such that $\gamma (b)-\gamma (a)$ is orthogonal to $\gamma (d)-\gamma (c)$ whenever $0 \leq a \leq b \leq c \leq d \leq 1$. ($\gamma$ may be called a "curve with orthogonal increments".) Hint: Take $H = L^{2}$, and consider characteristic functions of certain subsets of [0,1].

Define $u_{s}(t) = e^{i s t}$ for all $s \in \mathbb {R}^{1}, t \in \mathbb {R}^{1}$. Let $X$ be the complex vector space consisting of all finite linear combinations of these functions $u_{s}$. If $f \in X$ and $g \in X$, show that

exists. Show that this inner product makes $X$ into a unitary space whose completion is a nonseparable Hilbert space $H$. Show also that $\left\{ u_{s}: s \in \mathbb {R}^{1}\right\}$ is a maximal orthonormal set in $H$.

Fix a positive integer $N$, put $\omega = e^{2 \pi i / N}$, prove the orthogonality relations

and use them to derive the identities

that hold in every inner product space if $N \geq 3$. Show also that