Orthonormal Sets of Vectors and Bases

Verify the statements in Example 4.3: let $\mathcal{H} = L_{\mathbb {C}}^2[0, 2\pi ]$ and for $n$ in $\mathbb {Z}$ define $e_n$ in $\mathcal{H}$ by $e_n(t) = (2\pi )^{-1/2} \exp (int)$. Then $\left\{ e_n : n \in \mathbb {Z}\right\}$ is an orthonormal set in $\mathcal{H}$.

Verify the statements in Example 4.4: if $\mathcal{H} = \mathbb {F}^d$ and for $1 \leq k \leq d$, $e_k =$ the $d$-tuple with $1$ in the $k$th place and zeros elsewhere, then $\left\{ e_1, \ldots , e_d\right\}$ is a basis for $\mathcal{H}$.

Verify the statements in Example 4.5: let $\mathcal{H} = l^2(I)$ as in Example 1.7. For each $i$ in $I$ define $e_i$ in $\mathcal{H}$ by $e_i(i) = 1$ and $e_i(j) = 0$ for $j \neq i$. Then $\left\{ e_i : i \in I\right\}$ is a basis.

Find an infinite orthonormal set in the Hilbert space of Example 1.8 (the collection of all absolutely continuous functions $f : [0,1] \to \mathbb {F}$ with $f(0) = 0$ and $f' \in L^2(0,1)$, with $\left\langle f, g \right\rangle = \int_0^1 f'(t)\overline{g'(t)} \, dt$).

Using the notation of the Gram--Schmidt Orthogonalization Process, show that up to scalar multiple $e_1 = h_1/\left\| h_1\right\|$ and for $n \geq 2$, $e_n = \left\| h_n - f_n\right\|^{-1}(h_n - f_n)$, where $f_n$ is the vector defined formally by

If the sequence $1, x, x^2, \ldots$ is orthogonalized in $L^2(-1,1)$, the sequence $e_n(x) = \left[\frac{1}{2}(2n+1)\right]^{1/2} P_n(x)$ is obtained, where

The functions $P_n(x)$ are called Legendre polynomials.

If the sequence $e^{-x^2/2}, xe^{-x^2/2}, x^2 e^{-x^2/2}, \ldots$ is orthogonalized in $L^2(-\infty , \infty )$, the sequence $e_n(x) = [2^n n! \sqrt{\pi }]^{-1/2} H_n(x) e^{-x^2/2}$ is obtained, where

The functions $H_n$ are Hermite polynomials and satisfy $H_n'(x) = 2n H_{n-1}(x)$.

If the sequence $e^{-x/2}, xe^{-x/2}, x^2 e^{-x/2}, \ldots$ is orthogonalized in $L^2(0, \infty )$, the sequence $e_n(x) = e^{-x/2} L_n(x)/n!$ is obtained, where

The functions $L_n$ are called Laguerre polynomials.

Prove Corollary 4.10: if $\mathcal{E}$ is an orthonormal set and $h \in \mathcal{H}$, then $\sum_{e \in \mathcal{E}} \left| \left\langle h, e \right\rangle \right|^2 \leq \left\| h\right\|^2$, where the sum is understood in the sense of Definition 4.11 (that is, as the limit of the net $h_F = \sum \left\{ h_i : i \in F\right\}$ indexed by finite subsets $F$ of the index set).

If $\left\{ h_n\right\}$ is a sequence in Hilbert space and $\sum \left\{ h_n : n \in \mathbb {N}\right\}$ converges to $h$ (in the sense of Definition 4.11, i.e., as the limit of the net of finite partial sums indexed by finite subsets), then $\lim_n \sum_{k=1}^{n} h_k = h$. Show that the converse is false.

If $\left\{ h_n\right\}$ is a sequence in a Hilbert space and $\sum_{n=1}^{\infty } \left\| h_n\right\| < \infty$, show that $\sum \left\{ h_n : n \in \mathbb {N}\right\}$ converges in the sense of Definition 4.11.

Let $\left\{ \alpha_n\right\}$ be a sequence in $\mathbb {F}$ and prove that the following statements are equivalent:

1. $\sum \left\{ \alpha_n : n \in \mathbb {N}\right\}$ converges in the sense of Definition 4.11.

2. If $\pi$ is any permutation of $\mathbb {N}$, then $\sum_{n=1}^{\infty } \alpha_{\pi (n)}$ converges (unconditional convergence).

3. $\sum_{n=1}^{\infty } \left|\alpha_n\right| < \infty$.

Let $\mathcal{E}$ be an orthonormal subset of $\mathcal{H}$ and let $\mathcal{M} = \bigvee \mathcal{E}$. If $P$ is the orthogonal projection of $\mathcal{H}$ onto $\mathcal{M}$, show that $Ph = \sum \left\{ \left\langle h, e \right\rangle e : e \in \mathcal{E}\right\}$ for every $h$ in $\mathcal{H}$.

Let $\lambda =$ Area measure on $\left\{ z \in \mathbb {C} : \left|z\right| < 1\right\}$ and show that $1, z, z^2, \ldots$ are orthogonal vectors in $L^2(\lambda )$. Find $\left\| z^n\right\|$, $n \geq 0$. If $e_n = \left\| z^n\right\|^{-1} z^n$, $n \geq 0$, is $\left\{ e_0, e_1, \ldots \right\}$ a basis for $L^2(\lambda )$?

Proposition 4.14 states: if $\mathcal{H}$ is a Hilbert space, any two bases have the same cardinality. In the proof, letting $\mathcal{E}$ and $\mathcal{F}$ be two bases for $\mathcal{H}$ with $\varepsilon =$ the cardinality of $\mathcal{E}$ and $\eta =$ the cardinality of $\mathcal{F}$, show that if either $\varepsilon$ or $\eta$ is finite, then $\varepsilon = \eta$.

If $\mathcal{H}$ is an infinite dimensional Hilbert space, show that no orthonormal basis for $\mathcal{H}$ is a Hamel basis (a maximal linearly independent set). Show that a Hamel basis is uncountable.

Let $d \geq 1$ and let $\mu$ be a regular Borel measure on $\mathbb {R}^{d}$. Show that $L^2(\mu )$ is separable.

Suppose $L^2(X, \Omega , \mu )$ is separable and $\left\{ E_i : i \in I\right\}$ is a collection of pairwise disjoint subsets of $X$, $E_i \in \Omega$, and $0 < \mu (E_i) < \infty$ for all $i$. Show that $I$ is countable. Can you allow $\mu (E_i) = \infty$?

If $\left\{ h \in \mathcal{H} : \left\| h\right\| \leq 1\right\}$ is compact, show that $\dim \mathcal{H} < \infty$.

What is the cardinality of a Hamel basis for $l^2$?