107 lines
2.9 KiB
Markdown
107 lines
2.9 KiB
Markdown
# Topic 3: Separable Hilbert spaces
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## Infinite-dimensional Hilbert spaces
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Recall from Topic 1.
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[$L^2$ space](https://notenextra.trance-0.com/Math401/Math401_T1#section-3-further-definitions-in-measure-theory-and-integration)
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Let $\lambda$ be a measure on $\mathbb{R}$, or any other field you are interested in.
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A function is square integrable if
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$$
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\int_\mathbb{R} |f(x)|^2 d\lambda(x)<\infty
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$$
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### $L^2$ space and general Hilbert spaces
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#### Definition of $L^2(\mathbb{R},\lambda)$
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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).
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The Hermitian inner product is defined by
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$$
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\langle f,g\rangle=\int_\mathbb{R} \overline{f(x)}g(x) d\lambda(x)
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$$
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The norm is defined by
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$$
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\|f\|=\sqrt{\int_\mathbb{R} |f(x)|^2 d\lambda(x)}
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$$
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The space $L^2(\mathbb{R},\lambda)$ is complete.
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[Proof ignored here]
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> Recall the definition of [complete metric space](https://notenextra.trance-0.com/Math4111/Math4111_L17#definition-312).
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The inner product space $L^2(\mathbb{R},\lambda)$ is complete.
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#### Definition of general Hilbert space
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A Hilbert space is a complete inner product vector space.
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#### General Pythagorean theorem
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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 $v\in \mathscr{V}$,
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$$
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\|v\|^2=\sum_{i=1}^N |\langle v,u_i\rangle|^2+\left\|v-\sum_{i=1}^N \langle v,u_i\rangle u_i\right\|^2
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$$
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[Proof ignored here]
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#### Bessel's inequality
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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 $v\in \mathscr{V}$,
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$$
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\sum_{i=1}^N |\langle v,u_i\rangle|^2\leq \|v\|^2
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$$
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Immediate from the general Pythagorean theorem.
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### Orthonormal bases
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An orthonormal subset $S$ of a Hilbert space $\mathscr{H}$ is a set all of whose elements have norm 1 and are mutually orthogonal. ($\forall u,v\in S, \langle u,v\rangle=0$)
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#### Definition of orthonormal basis
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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.
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#### Theorem of existence of orthonormal basis
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Every separable Hilbert space has an orthonormal basis.
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[Proof ignored here]
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#### Theorem of Fourier series
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Let $\mathscr{H}$ be a separable Hilbert space with an orthonormal basis $\{e_n\}$. Then for any $f\in \mathscr{H}$,
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$$
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f=\sum_{n=1}^\infty \langle f,e_n\rangle e_n
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$$
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The series converges to some $g\in \mathscr{H}$.
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[Proof ignored here]
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#### Fourier series in $L^2([0,2\pi],\lambda)$
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Let $f\in L^2([0,2\pi],\lambda)$.
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$$
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f_N(x)=\sum_{n:|n|\leq N} c_n\frac{e^{inx}}{\sqrt{2\pi}}
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$$
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where $c_n=\frac{1}{2\pi}\int_0^{2\pi} f(x)e^{-inx} dx$.
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The series converges to some $f\in L^2([0,2\pi],\lambda)$ as $N\to \infty$.
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This is the Fourier series of $f$.
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