352 lines
12 KiB
Markdown
352 lines
12 KiB
Markdown
# Math401 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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> Note that **by some general result in point-set topology**, a normed vector space can always be enlarged so as to become complete. This process is called completion of the normed space.
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>
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> Some exercise is showing some hints for this result:
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>
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> Show that the subspace of $L^2(\mathbb{R},\lambda)$ consisting of square integrable continuous functions is not closed.
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>
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> Suggestion: consider the sequence of continuous functions $f_1(x), f_2(x),\cdots$, where $f_n(x)$ is defined by the following graph:
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>
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> 
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>
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> Show that $f_n$ converges in the $L^2$ norm to a function $f\in L^2(\mathbb{R},\lambda)$ but the limit function $f$ is not continuous. Draw the graph of $f_n$ to make this clear.
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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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#### Hermite polynomials
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The subspace spanned by polynomials is dense in $L^2(\mathbb{R},\lambda)$.
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An orthonormal basis of $L^2(\mathbb{R},\lambda)$ can be obtained by the Gram-Schmidt process on $\{1,x,x^2,\cdots\}$.
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The polynomials are called the Hermite polynomials.
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### Isomorphism and $\ell_2$ space
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#### Definition of isomorphic Hilbert spaces
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Let $\mathscr{H}_1$ and $\mathscr{H}_2$ be two Hilbert spaces.
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$\mathscr{H}_1$ 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.
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$$
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\langle Uf,Ug\rangle=\langle f,g\rangle
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$$
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for all $f,g\in \mathscr{H}_1$.
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When $\mathscr{H}_1=\mathscr{H}_2$, the map $U$ is called unitary.
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#### $\ell_2$ space
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The space $\ell_2$ is the space of all square summable sequences.
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$$
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\ell_2=\left\{(a_n)_{n=1}^\infty: \sum_{n=1}^\infty |a_n|^2<\infty\right\}
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$$
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An example of element in $\ell_2$ is $(1,0,0,\cdots)$.
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With inner product
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$$
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\langle (a_n)_{n=1}^\infty, (b_n)_{n=1}^\infty\rangle=\sum_{n=1}^\infty \overline{a_n}b_n
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$$
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It is a Hilbert space (every Cauchy sequence in $\ell_2$ converges to some element in $\ell_2$).
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### Bounded operators and continuity
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Let $T:\mathscr{V}\to \mathscr{W}$ be a linear map between two vector spaces $\mathscr{V}$ and $\mathscr{W}$.
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We define the norm of $\|\cdot\|$ on $\mathscr{V}$ and $\mathscr{W}$.
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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 $\mathscr{W}$.
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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 $\|T(u)-T(v)\|<\epsilon$.
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#### Definition of bounded operator
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A linear map $T:\mathscr{V}\to \mathscr{W}$ is bounded if
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$$
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\|T\|=\sup_{\|u\|=1}\|T(u)\|< \infty
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$$
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#### Theorem of continuity and boundedness
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A linear map $T:\mathscr{V}\to \mathscr{W}$ is continuous if and only if it is bounded.
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[Proof ignored here]
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#### Definition of bounded Hilbert space
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The set of all bounded linear operators in $\mathscr{V}$ is denoted by $\mathscr{B}(\mathscr{V})$.
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### Direct sum of Hilbert spaces
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Suppose $\mathscr{H}_1$ and $\mathscr{H}_2$ are two Hilbert spaces.
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The direct sum of $\mathscr{H}_1$ and $\mathscr{H}_2$ is the Hilbert space $\mathscr{H}_1\oplus \mathscr{H}_2$ with the inner product
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$$
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\langle (u_1,u_2),(v_1,v_2)\rangle=\langle u_1,v_1\rangle_{\mathscr{H}_1}+\langle u_2,v_2\rangle_{\mathscr{H}_2}
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$$
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Such space is denoted by $\mathscr{H}_1\oplus \mathscr{H}_2$.
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A countable direct sum of Hilbert spaces can be defined similarly, as long as it is bounded.
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That is, $\{u_n:n=1,2,\cdots\}$ is a sequence of elements in $\mathscr{H}_n$, and $\sum_{n=1}^\infty \|u_n\|^2<\infty$.
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The inner product in such countable direct sum is defined by
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$$
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\langle (u_n)_{n=1}^\infty, (v_n)_{n=1}^\infty\rangle=\sum_{n=1}^\infty \langle u_n,v_n\rangle_{\mathscr{H}_n}
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$$
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Such space is denoted by $\mathscr{H}=\bigoplus_{n=1}^\infty \mathscr{H}_n$.
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### Closed subspaces of Hilbert spaces
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#### Definition of closed subspace
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A subspace $\mathscr{M}$ of a Hilbert space $\mathscr{H}$ is closed if every convergent sequence in $\mathscr{M}$ converges to some element in $\mathscr{M}$.
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#### Definition of pairwise orthogonal subspaces
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Two subspaces $\mathscr{M}_1$ and $\mathscr{M}_2$ of a Hilbert space $\mathscr{H}$ are pairwise orthogonal if $\langle u,v\rangle=0$ for all $u\in \mathscr{M}_1$ and $v\in \mathscr{M}_2$.
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### Orthogonal projections
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#### Definition of orthogonal complement
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The orthogonal complement of a subspace $\mathscr{M}$ of a Hilbert space $\mathscr{H}$ is the set of all elements in $\mathscr{H}$ that are orthogonal to every element in $\mathscr{M}$.
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It is denoted by $\mathscr{M}^\perp=\{u\in \mathscr{H}: \langle u,v\rangle=0,\forall v\in \mathscr{M}\}$.
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#### Projection theorem
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Let $\mathscr{H}$ be a Hilbert space and $\mathscr{M}$ be a closed subspace of $\mathscr{H}$. Then for any $v\in \mathscr{H}$ can be uniquely decomposed as $v=u+w$ where $u\in \mathscr{M}$ and $w\in \mathscr{M}^\perp$.
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[Proof ignored here]
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### Dual Hilbert spaces
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#### Norm of linear functionals
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Let $\mathscr{H}$ be a Hilbert space.
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The norm of a linear functional $f\in \mathscr{H}^*$ is defined by
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$$
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\|f\|=\sup_{\|u\|=1}|f(u)|
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$$
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#### Definition of dual Hilbert space
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The dual Hilbert space of $\mathscr{H}$ is the space of all bounded linear functionals on $\mathscr{H}$.
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It is denoted by $\mathscr{H}^*$.
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$$
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\mathscr{H}^*=\mathscr{B}(\mathscr{H},\mathbb{C})=\{f: \mathscr{H}\to \mathbb{C}: f\text{ is linear and }\|f\|<\infty\}
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$$
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You can exchange the $\mathbb{C}$ with any other field you are interested in.
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#### The Riesz lemma
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For each $f\in \mathscr{H}^*$, there exists a unique $v_f\in \mathscr{H}$ such that $f(u)=\langle u,v_f\rangle$ for all $u\in \mathscr{H}$. And $\|f\|=\|v_f\|$.
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[Proof ignored here]
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#### Definition of bounded sesqilinear form
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A bounded sesqilinear form on $\mathscr{H}$ is a function $B: \mathscr{H}\times \mathscr{H}\to \mathbb{C}$ satisfying
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1. $B(u,av+bw)=aB(u,v)+bB(u,w)$ for all $u,v,w\in \mathscr{H}$ and $a,b\in \mathbb{C}$.
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2. $B(av+bw,u)=\overline{a}B(v,u)+\overline{b}B(w,u)$ for all $u,v,w\in \mathscr{H}$ and $a,b\in \mathbb{C}$.
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3. $|B(u,v)|\leq C\|u\|\|v\|$ for all $u,v\in \mathscr{H}$ and some constant $C>0$.
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There exists a unique bounded linear operator $A\in \mathscr{B}(\mathscr{H})$ such that $B(u,v)=\langle Au,v\rangle$ for all $u,v\in \mathscr{H}$. The norm of $A$ is the smallest constant $C$ such that $|B(u,v)|\leq C\|u\|\|v\|$ for all $u,v\in \mathscr{H}$.
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[Proof ignored here]
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### The adjoint of a bounded operator
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Let $A\in \mathscr{B}(\mathscr{H})$. And bounded sesqilinear form $B: \mathscr{H}\times \mathscr{H}\to \mathbb{C}$ such that $B(u,v)=\langle u,Av\rangle$ for all $u,v\in \mathscr{H}$. Then there exists a unique bounded linear operator $A^*\in \mathscr{B}(\mathscr{H})$ such that $B(u,v)=\langle A^*u,v\rangle$ for all $u,v\in \mathscr{H}$.
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[Proof ignored here]
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And $\|A^*\|=\|A\|$.
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Additional properties of bounded operators:
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Let $A,B\in \mathscr{B}(\mathscr{H})$ and $a,b\in \mathbb{C}$. Then
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1. $(aA+bB)^*=\overline{a}A^*+\overline{b}B^*$.
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2. $(AB)^*=B^*A^*$.
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3. $(A^*)^*=A$.
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4. $\|A^*\|=\|A\|$.
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5. $\|A^*A\|=\|A\|^2$.
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#### Definition of self-adjoint operator
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An operator $A\in \mathscr{B}(\mathscr{H})$ is self-adjoint if $A^*=A$.
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#### Definition of normal operator
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An operator $N\in \mathscr{B}(\mathscr{H})$ is normal if $NN^*=N^*N$.
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#### Definition of unitary operator
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An operator $U\in \mathscr{B}(\mathscr{H})$ is unitary if $U^*U=UU^*=I$.
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where $I$ is the identity operator on $\mathscr{H}$.
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#### Definition of orthogonal projection
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An operator $P\in \mathscr{B}(\mathscr{H})$ is an orthogonal projection if $P^*=P$ and $P^2=P$.
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### Tensor product of (infinite-dimensional) Hilbert spaces
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#### Definition of tensor product
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Let $\mathscr{H}_1$ and $\mathscr{H}_2$ be two Hilbert spaces. $u_1\in \mathscr{H}_1$ and $u_2\in \mathscr{H}_2$. Then $u_1\otimes u_2$ is an conjugate bilinear functional on $\mathscr{H}_1\times \mathscr{H}_2$.
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$$
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(u_1\otimes u_2)(v_1,v_2)=\langle u_1,v_1\rangle_{\mathscr{H}_1}\langle u_2,v_2\rangle_{\mathscr{H}_2}
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$$
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Let $\mathscr{V}$ be the set of all finite lienar combination of such conjugate bilinear functionals. We define the inner product on $\mathscr{V}$ by
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$$
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\langle u\otimes v,u'\otimes v'\rangle=\langle u,u'\rangle_{\mathscr{H}_1}\langle v,v'\rangle_{\mathscr{H}_2}
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$$
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The infinite-dimensional tensor product of $\mathscr{H}_1$ and $\mathscr{H}_2$ is the completion (extension of those bilinear functionals to make the set closed) of $\mathscr{V}$ with respect to the norm induced by the inner product.
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Denoted by $\mathscr{H}_1\otimes \mathscr{H}_2$.
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The orthonormal basis of $\mathscr{H}_1\otimes \mathscr{H}_2$ is $\{u_i\otimes v_j:i=1,2,\cdots,j=1,2,\cdots\}$. where $u_i$ is the orthonormal basis of $\mathscr{H}_1$ and $v_j$ is the orthonormal basis of $\mathscr{H}_2$.
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### Fock space
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#### Definition of Fock space
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Let $\mathscr{H}^{\otimes n}$ be the $n$-fold tensor product of $\mathscr{H}$.
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Set $\mathscr{H}^{\otimes 0}=\mathbb{C}$.
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The Fock space of $\mathscr{H}$ is the direct sum of all $\mathscr{H}^{\otimes n}$.
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$$
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\mathscr{F}(\mathscr{H})=\bigoplus_{n=0}^\infty \mathscr{H}^{\otimes n}
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$$
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For example, if $\mathscr{H}=L^2(\mathbb{R},\lambda)$, then an element in $\mathscr{F}(\mathscr{H})$ is a sequence of functions $\psi=(\psi_0,\psi_1(x_1),\psi_2(x_1,x_2),\cdots)$ such that $|\psi_0|^2+\sum_{n=1}^\infty \int|\psi_n(x_1,\cdots,x_n)|^2dx_1\cdots dx_n<\infty$.
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