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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.