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Topic 3: Separable Hilbert spaces
Infinite-dimensional Hilbert spaces
Recall from Topic 1.
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.
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}).