176 lines
5.0 KiB
Markdown
176 lines
5.0 KiB
Markdown
# Topic 3: Separable Hilbert spaces
|
|
|
|
## Infinite-dimensional Hilbert spaces
|
|
|
|
Recall from Topic 1.
|
|
|
|
[$L^2$ space](https://notenextra.trance-0.com/Math401/Math401_T1#section-3-further-definitions-in-measure-theory-and-integration)
|
|
|
|
Let $\lambda$ be a measure on $\mathbb{R}$, or any other field you are interested in.
|
|
|
|
A function is square integrable if
|
|
|
|
$$
|
|
\int_\mathbb{R} |f(x)|^2 d\lambda(x)<\infty
|
|
$$
|
|
|
|
### $L^2$ space and general Hilbert spaces
|
|
|
|
#### Definition of $L^2(\mathbb{R},\lambda)$
|
|
|
|
The space $L^2(\mathbb{R},\lambda)$ is the space of all square integrable, measurable functions on $\mathbb{R}$ with respect to the measure $\lambda$ (The Lebesgue measure).
|
|
|
|
The Hermitian inner product is defined by
|
|
|
|
$$
|
|
\langle f,g\rangle=\int_\mathbb{R} \overline{f(x)}g(x) d\lambda(x)
|
|
$$
|
|
|
|
The norm is defined by
|
|
|
|
$$
|
|
\|f\|=\sqrt{\int_\mathbb{R} |f(x)|^2 d\lambda(x)}
|
|
$$
|
|
|
|
The space $L^2(\mathbb{R},\lambda)$ is complete.
|
|
|
|
[Proof ignored here]
|
|
|
|
> Recall the definition of [complete metric space](https://notenextra.trance-0.com/Math4111/Math4111_L17#definition-312).
|
|
|
|
The inner product space $L^2(\mathbb{R},\lambda)$ is complete.
|
|
|
|
#### Definition of general Hilbert space
|
|
|
|
A Hilbert space is a complete inner product vector space.
|
|
|
|
#### General Pythagorean theorem
|
|
|
|
Let $u_1,u_2,\cdots,u_N$ be an orthonormal set in an inner product space $\mathscr{V}$ (may not be complete). Then for all $v\in \mathscr{V}$,
|
|
|
|
$$
|
|
\|v\|^2=\sum_{i=1}^N |\langle v,u_i\rangle|^2+\left\|v-\sum_{i=1}^N \langle v,u_i\rangle u_i\right\|^2
|
|
$$
|
|
|
|
[Proof ignored here]
|
|
|
|
#### Bessel's inequality
|
|
|
|
Let $u_1,u_2,\cdots,u_N$ be an orthonormal set in an inner product space $\mathscr{V}$ (may not be complete). Then for all $v\in \mathscr{V}$,
|
|
|
|
$$
|
|
\sum_{i=1}^N |\langle v,u_i\rangle|^2\leq \|v\|^2
|
|
$$
|
|
|
|
Immediate from the general Pythagorean theorem.
|
|
|
|
### Orthonormal bases
|
|
|
|
An orthonormal subset $S$ of a Hilbert space $\mathscr{H}$ is a set all of whose elements have norm 1 and are mutually orthogonal. ($\forall u,v\in S, \langle u,v\rangle=0$)
|
|
|
|
#### Definition of orthonormal basis
|
|
|
|
An orthonormal subset of $S$ of a Hilbert space $\mathscr{H}$ is an orthonormal basis of $\mathscr{H}$ if there are no other orthonormal subsets of $\mathscr{H}$ that contain $S$ as a proper subset.
|
|
|
|
#### Theorem of existence of orthonormal basis
|
|
|
|
Every separable Hilbert space has an orthonormal basis.
|
|
|
|
[Proof ignored here]
|
|
|
|
#### Theorem of Fourier series
|
|
|
|
Let $\mathscr{H}$ be a separable Hilbert space with an orthonormal basis $\{e_n\}$. Then for any $f\in \mathscr{H}$,
|
|
|
|
$$
|
|
f=\sum_{n=1}^\infty \langle f,e_n\rangle e_n
|
|
$$
|
|
|
|
The series converges to some $g\in \mathscr{H}$.
|
|
|
|
[Proof ignored here]
|
|
|
|
#### Fourier series in $L^2([0,2\pi],\lambda)$
|
|
|
|
Let $f\in L^2([0,2\pi],\lambda)$.
|
|
|
|
$$
|
|
f_N(x)=\sum_{n:|n|\leq N} c_n\frac{e^{inx}}{\sqrt{2\pi}}
|
|
$$
|
|
|
|
where $c_n=\frac{1}{2\pi}\int_0^{2\pi} f(x)e^{-inx} dx$.
|
|
|
|
The series converges to some $f\in L^2([0,2\pi],\lambda)$ as $N\to \infty$.
|
|
|
|
This is the Fourier series of $f$.
|
|
|
|
#### Hermite polynomials
|
|
|
|
The subspace spanned by polynomials is dense in $L^2(\mathbb{R},\lambda)$.
|
|
|
|
An orthonormal basis of $L^2(\mathbb{R},\lambda)$ can be obtained by the Gram-Schmidt process on $\{1,x,x^2,\cdots\}$.
|
|
|
|
The polynomials are called the Hermite polynomials.
|
|
|
|
### Isomorphism and $\ell_2$ space
|
|
|
|
#### Definition of isomorphic Hilbert spaces
|
|
|
|
Let $\mathscr{H}_1$ and $\mathscr{H}_2$ be two Hilbert spaces.
|
|
|
|
$\mathscr{H}_1$ and $\mathscr{H}_2$ are isomorphic if there exists a surjective linear map $U:\mathscr{H}_1\to \mathscr{H}_2$ that is bijective and preserves the inner product.
|
|
|
|
$$
|
|
\langle Uf,Ug\rangle=\langle f,g\rangle
|
|
$$
|
|
|
|
for all $f,g\in \mathscr{H}_1$.
|
|
|
|
When $\mathscr{H}_1=\mathscr{H}_2$, the map $U$ is called unitary.
|
|
|
|
#### $\ell_2$ space
|
|
|
|
The space $\ell_2$ is the space of all square summable sequences.
|
|
|
|
$$
|
|
\ell_2=\left\{(a_n)_{n=1}^\infty: \sum_{n=1}^\infty |a_n|^2<\infty\right\}
|
|
$$
|
|
|
|
An example of element in $\ell_2$ is $(1,0,0,\cdots)$.
|
|
|
|
With inner product
|
|
|
|
$$
|
|
\langle (a_n)_{n=1}^\infty, (b_n)_{n=1}^\infty\rangle=\sum_{n=1}^\infty \overline{a_n}b_n
|
|
$$
|
|
|
|
It is a Hilbert space (every Cauchy sequence in $\ell_2$ converges to some element in $\ell_2$).
|
|
|
|
### Bounded operators and continuity
|
|
|
|
Let $T:\mathscr{V}\to \mathscr{W}$ be a linear map between two vector spaces $\mathscr{V}$ and $\mathscr{W}$.
|
|
|
|
We define the norm of $\|\cdot\|$ on $\mathscr{V}$ and $\mathscr{W}$.
|
|
|
|
Then $T$ is continuous if for all $u\in \mathscr{V}$, if $u_n\to u$ in $\mathscr{V}$, then $T(u_n)\to T(u)$ in $\mathscr{W}$.
|
|
|
|
Using the delta-epsilon language, we can say that $T$ is continuous if for all $\epsilon>0$, there exists a $\delta>0$ such that if $\|u-v\|<\delta$, then $\|T(u)-T(v)\|<\epsilon$.
|
|
|
|
#### Definition of bounded operator
|
|
|
|
A linear map $T:\mathscr{V}\to \mathscr{W}$ is bounded if
|
|
|
|
$$
|
|
\|T\|=\sup_{\|u\|=1}\|T(u)\|< \infty
|
|
$$
|
|
|
|
#### Theorem of continuity and boundedness
|
|
|
|
A linear map $T:\mathscr{V}\to \mathscr{W}$ is continuous if and only if it is bounded.
|
|
|
|
[Proof ignored here]
|
|
|
|
#### Definition of bounded Hilbert space
|
|
|
|
The set of all bounded linear operators in $\mathscr{V}$ is denoted by $\mathscr{B}(\mathscr{V})$.
|