13.2 Expansion theorems

中文

Subspaces

A subspace $V$ of a Hilbert space $\mathcal{H}$ is a vector subspace that is closed with respect to the norm topology. For a vector subspace to be closed we require the limit of any sequence of vectors in $V$ to belong to $V$,

$$u_1,u_2,\cdots \to u\text{ and all }{u}_n\in V⟹u\in V.$$

If $V$ is any vector subspace of $\mathcal{H},$ , its closure $\overline{V}$ is the smallest subspace containing $V$. It is the intersection of all subspaces containing $V$.

If $K$ is any subset of $\mathcal{H}$ then, as in Chapter 3, the vector subspace generated by $K$ is

$$L(K)=\left\{\sum _{i=1}^n{\alpha }^i{u}_i|{\alpha }^i\in \mathbb{C},u_i\in K\right\},$$

but the subspace generated by $K$ will always refer to the closed subspace $\overline{L(K)}$ generated by $K$. A Hilbert space is called separable if there is a countable set $K=\{u_1,u_2,...\}$ such that $\mathcal{H}$ is generated by $K$,

$$\mathcal{H}=\overline{L(K)}=\overline{L(u_1,u_2,\dots )}.$$

Orthonormal bases

If the Hilbert space $\mathcal{H}$ is separable and is generated by $\{u_1,u_2,\dots \}$ , we may use the Schmidt orthonormalization procedure (see Section 5.2) to produce an orthonormal set $\{e_1,e_2,\dots ,e_n\}$

$$⟨e_i|e_j⟩={\delta }_{ij}=\{\begin{array}{ll}1& \text{ if }i=j,\\ 0& \text{ if }i\ne j.\end{array}$$

The steps of the procedure are

$$\begin{array}{ll}{f}_1=u_1& e_1=f_1/\Vert f_1\Vert \\ f_2=u_2-⟨e_1|u_2⟩e_1& e_2=f_2/\Vert f_2\Vert \\ f_3=u_3-⟨e_1|u_3⟩e_1-⟨e_2|u_3⟩e_2& e_3=f_3/\Vert f_3\Vert ,\text{e t c .}\end{array}$$

from which it can be seen that each $u_n$ is a linear combination of $\{e_1,e_2,\dots ,e_n\}$ . Hence $\mathcal{H}=\overline{L(\{e_1,e_2,\dots \})}$ and the set $\{e_n\mid n=1,2,\dots \}$ is called a complete orthonormal set or orthonormal basis of $\mathcal{H}.$

Theorem 13.2 · Expansion in an orthonormal basis

If $\mathcal{H}$ is a separable Hilbert space and $\{e_1,e_2,\dots \}$ is a complete orthonormal set, then any vector $u\in \mathcal{H}$ has a unique expansion

$$u=\sum _{n=1}^{\infty }{c}_n{e}_n\text{where}{c}_n=⟨e_n|u⟩.\text{\hspace{1em}(13.4)}$$

The meaning of the sum in this theorem is

$$\Vert u-\sum _{n=1}^N{c}_n{e}_n\Vert \to 0\text{ as }N\to \infty .$$

A critical part of the proof is Bessel’s inequality:

$$\sum _{n=1}^N{\left|⟨e_n|u⟩\right|}^2\le \Vert u{\Vert }^2.\text{\hspace{1em}(13.5)}$$

Proof

For any $N>1$

$$\begin{array}{l}0\le {\Vert u-{\sum }_{n=1}^N⟨e_n|u⟩e_n\Vert }^2\\ =⟨u-{\sum }_n⟨e_n|u⟩e_n|u-{\sum }_m⟨e_m|u⟩e_m⟩\\ =\Vert u{\Vert }^2-2{\sum }_{n=1}^N\overline{⟨e_n|u⟩}⟨e_n|u⟩+{\sum }_{n=1}^N{\sum }_{m=1}^N\overline{⟨e_n|u⟩}{\delta }_{mn}⟨e_m|u⟩\\ =\Vert u{\Vert }^2-{\sum }_{n=1}^N|⟨e_n|u⟩{|}^2,\end{array}$$

which gives the desired inequality.

Taking the limit $N\to \infty$ in Bessel’s inequality Eq. (13.5) shows that the series

$$\sum _{n=1}^{\infty }{\left|⟨e_n|u⟩\right|}^2$$

is bounded above and therefore convergent since it consists entirely of non-negative terms. To prove the expansion theorem 13.2, we first show two lemmas.

Lemma 13.3 · Continuity of the inner product in one variable

If $v_n\to v$ in a Hilbert space $\mathcal{H},$ then for all vectors $u\in \mathcal{H}$

$$⟨u\mid v_n⟩\to ⟨u\mid v⟩.$$

Proof

By the Cauchy–Schwarz inequality (5.13)

$$\begin{array}{l}|⟨u|v_n⟩-⟨u|v⟩|=|⟨u|v_n-v⟩|\\ \le \Vert u\Vert \Vert v_n-v\Vert \\ \to 0\end{array}$$

Lemma 13.4 · Orthogonality criterion for a complete orthonormal set

If $\{e_1,e_2,\dots \}$ is a complete orthonormal set and $⟨v|e_n⟩=0$ for $n=$ 1, 2, … then $v=0$

Proof

Since $\{e_n\}$ is a complete o.n. set, every vector $v\in \mathcal{H}$ is the limit of a sequence of vectors spanned by the vectors $\{e_1,e_2,\dots \}$

$$v=\underset{n\to \infty }{\lim }{v}_n\text{ where }{v}_n=\sum _{i=1}^N{v}_{ni}{e}_i.$$

Setting $u=v$ in Lemma 13.3, we have

$$\Vert v{\Vert }^2=⟨v|v⟩=\underset{n\to \infty }{\lim }⟨v|v_n⟩=0.$$

Hence $v=0$ by the condition (Norm1).

We now return to the proof of the expansion theorem.

Proof of Theorem 13.2

Set

$$u_N=\sum _{n=1}^N⟨e_n|u⟩e_n.$$

This is a Cauchy sequence,

$${\Vert u_N-u_M\Vert }^2=\sum _{n=M}^N{\left|⟨e_n\mid u⟩\right|}^2\to 0\text{ as }M,N\to \infty$$

since the series ${\sum }_n{\left|⟨e_n\mid u⟩\right|}^2$ is absolutely convergent by Bessel’s inequality Eq. (13.5). By completeness of the Hilbert space $\mathcal{H}$, $u_N\to u^{\prime }$ for some vector $u^{\prime }\in \mathcal{H}.$ . But

$$⟨e_k|u-u^{\prime }⟩=\underset{N\to \infty }{\lim }⟨e_k|u-u_N⟩=⟨e_k|u⟩-e_{k}u=0$$

since $⟨e_k\mid u_N⟩=⟨e_k\mid u⟩$ for all $N\ge k$ . Hence, by Lemma 13.4,

$$u=u^{\prime }=\underset{N\to \infty }{\lim }{u}_N,$$

and Theorem 13.2 is proved.

Exercise

Show that every separable Hilbert space is either a finite dimensional inner product space, or is isomorphic with ${\ell }^2$ .

Example 13.5

For any real numbers $a<b$ the Hilbert space $L^2([a,b])$ is separable. The following is an outline proof; details may be found in [1]. By Theorem 11.2 any posi tive measurable function $f\ge 0$ on $[a,b]$ may be approximated by an increasing sequence of positive simple functions $0<s_n(x)f(x)$ . If $f\in {\mathcal{L}}^2([a,b])$ then by the dominated convergence, Theorem 11.11, $\Vert f-s_n\Vert \to 0$ . By a straightforward, but slightly techni cal, argument these simple functions may be approximated with continuous functions, and prove that for any $ϵ>0$ there exists a positive continuous function $h(x)$ such that $\Vert f-h\Vert <ϵ$ . Using a famous theorem of Weierstrass that any continuous function on a closed interval can be arbitrarily closely approximated by polynomials, it is possible to find a complex-valued polynomial $p(x)$ such that $\Vert f-p\Vert <ϵ$ . Since all polynomials are of the form $p(x)=c_0+c_{1}x+c_2{x}^2+\cdot \cdot \cdot +c_n{x}^n$ where $c\in \mathbb{C}.$ , the functions $1,x,x^2,\dots$ . form a countable sequence of functions on $[a,b]$ that generate $L^2([a,b])$ ). This proves separability of $L^2([a,b])$ .

Separability of $L^2(\mathbb{R})$ is proved by showing the restricted polynomial functions $f_{n,N}=$ ${\mathbf{x}}^n{\chi }_{[-N,N]}$ are a countable set that generates $L^2(\mathbb{R})$ .

Example 13.6

On $L^2([-\pi ,\pi ])$ the functions

$${\varphi }_n(x)=\frac{{\mathrm{e}}^{inx}}{\sqrt{2\pi }}$$

form an orthonormal basis,

$$⟨{\varphi }_m|{\varphi }_n⟩=\frac{1}{2\pi }{\int }_{-\pi }^{\pi }{\mathrm{e}}^{i(n-m)x}\mathrm{d}x={\delta }_{mn}$$

as is easily calculated for the two separate cases $n\ne m$ and $n=m$ . These generate the Fourier series of an arbitrary square integrable function $f$ on $[-\pi ,\pi ]$

$$f=\sum _{n=-\infty }^{\infty }{c}_n{\varphi }_{n}a.e.$$

where $c_n$ are the Fourier coefficients

$$c_n=⟨{\varphi }_n|f⟩=\frac{1}{\sqrt{2\pi }}{\int }_{-\pi }^{\pi }{\mathrm{e}}^{-inx}f(x)\mathrm{d}x.$$

Example 13.7

The Hermite polynomials $H_n(x)(n=0,1,2,...)$ are defined by

$$H_n(x)=(-1{)}^n{\mathrm{e}}^{x^2}\frac{{\mathrm{d}}^n{\mathrm{e}}^{-x^2}}{\mathrm{d}{x}^n}.$$

The first few are

$$H_0(x)=1,H_1(x)=2x,H_2(x)4x^2-2,H_3(x)=8x^3-12x,\dots$$

The $n$th polynomial is clearly ofdegree $n$ with leading term $(-2x{)}^n$ . The functions ${\psi }_n(x)=$ ${\mathsf{e}}^{-(1/2)x^2}{H}_n(x)$ form an orthogonal system in $L^2(\mathbb{R})$ :

$$\begin{array}{rl}⟨{\psi }_m|{\psi }_n⟩& =(-1{)}^{n+m}{\int }_{-\infty }^{\infty }{\mathrm{e}}^{x^2}\frac{{\mathrm{d}}^m{\mathrm{e}}^{-x^2}}{\mathrm{d}{x}^m}\frac{{\mathrm{d}}^n{\mathrm{e}}^{-x^2}}{\mathrm{d}{x}^n}\mathrm{d}x\\ & =(-1{)}^{n+m}\left({\left[{\mathrm{e}}^{x^2}\frac{{\mathrm{d}}^m{\mathrm{e}}^{-x^2}}{\mathrm{d}{x}^m}\frac{{\mathrm{d}}^{n-1}{\mathrm{e}}^{-x^2}}{\mathrm{d}{x}^{n-1}}\right]}_{-\infty }^{\infty }-{\int }_{-\infty }^{\infty }\frac{\mathrm{d}}{\mathrm{d}x}\left({\mathrm{e}}^{x^2}\frac{{\mathrm{d}}^m{\mathrm{e}}^{-x^2}}{\mathrm{d}{x}^m}\right)\frac{{\mathrm{d}}^{n-1}{\mathrm{e}}^{-x^2}}{\mathrm{d}{x}^{n-1}}\mathrm{d}x\right)\end{array}$$

on integration by parts. The first expression on the right-hand side of this equation vanishes since it involves terms of order $e^{-x^2}{x}^k$ that approach 0 as $x\pm \infty$ . We may repeat the integration by parts on the remaining integral, until we arrive at

$$⟨{\psi }_m|{\psi }_n⟩=(-1{)}^m{\int }_{-\infty }^{\infty }{\mathrm{e}}^{-x^2}\frac{{\mathrm{d}}^n}{\mathrm{d}{x}^n}\left({\mathrm{e}}^{x^2}\frac{{\mathrm{d}}^m{\mathrm{e}}^{-x^2}}{\mathrm{d}{x}^m}\right)\mathrm{d}x,$$

which vanishes if $n>m$ since the expression in the brackets is a polynomial of degree $m$. A similar argument for $n<m$ yields

$$⟨{\psi }_m|{\psi }_n⟩=0\text{ for }n\ne m.$$

For $n=m$ we have, from the leading term in the hermite polynomials,

$$\begin{array}{l}\Vert {\psi }_n{\Vert }^2=⟨{\psi }_n|{\psi }_n⟩=(-1{)}^n{\int }_{-\infty }^{\infty }{\mathrm{e}}^{-x^2}\frac{{\mathrm{d}}^n}{\mathrm{d}{x}^n}\left({\mathrm{e}}^{x^2}\frac{{\mathrm{d}}^n{\mathrm{e}}^{-x^2}}{\mathrm{d}{x}^n}\right)\mathrm{d}x\\ =(-1{)}^n{\int }_{-\infty }^{\infty }{\mathrm{e}}^{-x^2}\frac{{\mathrm{d}}^n}{\mathrm{d}{x}^n}((-2x{)}^n)\mathrm{d}x\\ =2^{n}n!{\int }_{-\infty }^{\infty }{\mathrm{e}}^{-x^2}\mathrm{d}x\\ =2^{n}n!\sqrt{\pi }.\end{array}$$

Thus the functions

$${\varphi }_n(x)=\frac{{\mathrm{e}}^{-(1/2)x^2}}{\sqrt{2^{n}n!\sqrt{\pi }}}{H}_n(x)\text{\hspace{1em}(13.6)}$$

form an orthonormal set. From Weierstrass’s theorem they form a complete o.n. basis fo $L^2(\mathbb{R})$

The following generalization of Lemma 13.3 is sometimes useful.

Lemma 13.5 · Joint continuity of the inner product

$H{u}_{n}u$ and $v_n\to v$ then $u_n\mid v_{n}u\mid v$

Proof

Using the Cauchy–Schwarz inequality (5.13)

$$\begin{array}{rl}\left|⟨u_n|v_n⟩-⟨u|v⟩\right|& =\left|⟨u_n|v_n⟩-⟨u_n|v⟩+⟨u_n|v⟩-⟨u|v⟩\right|\\ & \le \left|⟨u_n|v_n⟩-⟨u_n|v⟩\right|+\left|⟨u_n|v⟩-⟨u|v⟩\right|\\ & \le \Vert u_n\Vert \Vert v_n-v\Vert +\Vert u_n-u\Vert \Vert v\Vert \\ & \to \Vert u\Vert .0+0.\Vert v\Vert \to 0.\end{array}$$

Exercise

If $u_n\to u$ show that $\Vert u_n\Vert \Vert u\Vert$ , used in the last step of the above proof.

The following identity has widespread application in quantum mechanics.

Theorem 13.6 · Parseval’s identity

(Parseval’s identity)

$$⟨u\mid v⟩=\sum _{i=1}^{\infty }⟨u\mid e_i⟩⟨e_i\mid v⟩.\text{\hspace{1em}(13.7)}$$

Proof

Set

$$u_n=\sum _{i=1}^n⟨e_i\mid u⟩e_i\text{ and }{v}_n=\sum _{i=1}^n⟨e_i\mid v⟩e_i.$$

By Theorem 13.2, $u_n\to u$ and $v_n\to v\text{as}n\to \infty$ . Now using Lemma 13.5,

$$\begin{array}{l}⟨u|v⟩={\lim }_{n\to \infty }⟨u_n|v_n⟩\\ ={\lim }_{n\to \infty }{\sum }_{i=1}^n{\sum }_{j=1}^n\overline{⟨e_i|u⟩}⟨e_j|v⟩⟨e_i|e_j⟩\\ ={\lim }_{n\to \infty }{\sum }_{i=1}^n{\sum }_{j=1}^n⟨u|e_i⟩⟨e_j|v⟩{\delta }_{ij}\\ ={\lim }_{n\to \infty }{\sum }_{i=1}^n⟨u|e_i⟩⟨e_i|v⟩.\\ ={\sum }_{i=1}^{\infty }⟨u|e_i⟩⟨e_i|v⟩.\end{array}$$

For a function $\begin{array}{r}f(x)={\sum }_{n=-\infty }^{\infty }{c}_n{\varphi }_n\mathrm{on}[-\pi ,\pi ],\end{array}$ , where ${\varphi }_n(x)$ are the standard Fourier functions given in Example 13.6, Parseval’s identity becomes the well-known formula

$$\Vert f{\Vert }^2={\int }_{-\pi }^{\pi }|f(x){|}^{2}dx=\sum _{n=-\infty }^{\infty }|c_n{|}^2.$$

Problems

Problem 13.4

Show that a vector subspace is a closed subset of with respect to the norm topology iff the limit of every sequence of vectors in $V$ belongs to $V$

Problem 13.5

Let ${\ell }_0$ be the subset of ${\ell }^2$ consisting of sequences with only finitely many terms different from zero. Show that ${\ell }_0$ is a vector subspace of ${\ell }^2$ , but that it is not closed. What is its closure $\overline{{\ell }_0!}$

Problem 13.6

We say a sequence $\{x_n\}$ converges weakly to a point $x$ in a Hilbert space $\mathcal{H},$ , written $x_{n}x\mathrm{if}⟨x_n|y⟩\to ⟨x|y⟩$ for all $y\in \mathcal{H}$ . Show that every strongly convergent sequence, $\Vert x_n-x\Vert \to$ 0 is weakly convergent to $x$. In finite dimensional Hilbert spaces show that every weakly convergent sequence is strongly convergent.

Give an example where $x_n\to x$ but $\Vert x_n\Vert \nrightarrow \Vert x\Vert$ . Is it true in general that the weak limit of a sequence is unique?

Show that if $x_n\to x$ and $\Vert x_n\Vert \ne \Vert x\Vert$ then $x_n\nrightarrow x$

Problem 13.7

In the Hilbert space $L^2([-1,1])$ let $\{f_n(x)\}$ be the sequence of functions $1,x,x^2,\dots ,f_n(x)=x^n$

(a) Apply Schmidt orthonormalization to this sequence, writing down the first three polynomials so obtained.

(b) The $n$th Legendre polynomial $P_n(x)$ is defined as

$$P_n(x)=\frac{1}{2^{n}n!}\frac{{\mathrm{d}}^n}{\mathrm{d}{x}^n}{\left(x^2-1\right)}^n.$$

Prove that

$${\int }_{-1}^1{P}_m(x)P_n(x)\mathrm{d}x=\frac{2}{2n+1}{\delta }_{mn}.$$

(c) Show that the $n$th member of the o.n. sequence obtained in (a) is $\sqrt{n+\frac{1}{2}}{P}_n(x)$

Problem 13.8

Show that Schmidt orthonormalization in $L^2(\mathbb{R})$ , applied to the sequence offunction

$$f_n(x)=x^n{\mathrm{e}}^{-x^2/2},$$

leads to the normalized hermite functions Eq. (13.6) of Example 13.7.

Problem 13.9

Show that applying Schmidt orthonormalization in $L^2([0,\infty ])$ to the sequence of functions

$$f_n(x)=x^n{\mathrm{e}}^{-x/2}$$

leads to a normalized sequence of functions involving the Laguerre polynomials

$$L_n(x)={\mathrm{e}}^x\frac{{\mathrm{d}}^n}{\mathrm{d}{x}^n}\left(x^n{\mathrm{e}}^{-x}\right).$$