update

pages/Math401/Math401_T3.md

@@ -40,6 +40,18 @@ The space $L^2(\mathbb{R},\lambda)$ is complete.
 
 The inner product space $L^2(\mathbb{R},\lambda)$ is complete.
 
+> 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.
+>
+> Some exercise is showing some hints for this result:
+> 
+> Show that the subspace of $L^2(\mathbb{R},\lambda)$ consisting of square integrable continuous functions is not closed.   
+> 
+> Suggestion: consider the sequence of continuous functions $f_1(x), f_2(x),\cdots$, where $f_n(x)$ is defined by the following graph:
+> 
+> ![function.png](https://notenextra.trance-0.com/Math401/L2_square_integrable_problem.png)
+>
+> 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.
+
 #### Definition of general Hilbert space
 
 A Hilbert space is a complete inner product vector space.
@@ -173,3 +185,167 @@ A linear map $T:\mathscr{V}\to \mathscr{W}$ is continuous if and only if it is b
 #### Definition of bounded Hilbert space
 
 The set of all bounded linear operators in $\mathscr{V}$ is denoted by $\mathscr{B}(\mathscr{V})$.
+
+### Direct sum of Hilbert spaces
+
+Suppose $\mathscr{H}_1$ and $\mathscr{H}_2$ are two Hilbert spaces.
+
+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
+
+$$
+\langle (u_1,u_2),(v_1,v_2)\rangle=\langle u_1,v_1\rangle_{\mathscr{H}_1}+\langle u_2,v_2\rangle_{\mathscr{H}_2}
+$$
+
+Such space is denoted by $\mathscr{H}_1\oplus \mathscr{H}_2$.
+
+A countable direct sum of Hilbert spaces can be defined similarly, as long as it is bounded.
+
+That is, $\{u_n:n=1,2,\cdots\}$ is a sequence of elements in $\mathscr{H}_n$, and $\sum_{n=1}^\infty \|u_n\|^2<\infty$.
+
+The inner product in such countable direct sum is defined by
+
+$$
+\langle (u_n)_{n=1}^\infty, (v_n)_{n=1}^\infty\rangle=\sum_{n=1}^\infty \langle u_n,v_n\rangle_{\mathscr{H}_n}
+$$
+
+Such space is denoted by $\mathscr{H}=\bigoplus_{n=1}^\infty \mathscr{H}_n$.
+
+### Closed subspaces of Hilbert spaces
+
+#### Definition of closed subspace
+
+A subspace $\mathscr{M}$ of a Hilbert space $\mathscr{H}$ is closed if every convergent sequence in $\mathscr{M}$ converges to some element in $\mathscr{M}$.
+
+#### Definition of pairwise orthogonal subspaces
+
+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 $v\in \mathscr{M}_2$.
+
+### Orthogonal projections
+
+#### Definition of orthogonal complement
+
+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 $\mathscr{M}$.
+
+It is denoted by $\mathscr{M}^\perp=\{u\in \mathscr{H}: \langle u,v\rangle=0,\forall v\in \mathscr{M}\}$.
+
+#### Projection theorem
+
+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 $w\in \mathscr{M}^\perp$.
+
+[Proof ignored here]
+
+### Dual Hilbert spaces
+
+#### Norm of linear functionals
+
+Let $\mathscr{H}$ be a Hilbert space.
+
+The norm of a linear functional $f\in \mathscr{H}^*$ is defined by
+
+$$
+\|f\|=\sup_{\|u\|=1}|f(u)|
+$$
+
+#### Definition of dual Hilbert space
+
+The dual Hilbert space of $\mathscr{H}$ is the space of all bounded linear functionals on $\mathscr{H}$.
+
+It is denoted by $\mathscr{H}^*$.
+
+$$
+\mathscr{H}^*=\mathscr{B}(\mathscr{H},\mathbb{C})=\{f: \mathscr{H}\to \mathbb{C}: f\text{ is linear and }\|f\|<\infty\}
+$$
+
+You can exchange the $\mathbb{C}$ with any other field you are interested in.
+
+#### The Riesz lemma
+
+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 $\|f\|=\|v_f\|$.
+
+[Proof ignored here]
+
+#### Definition of bounded sesqilinear form
+
+A bounded sesqilinear form on $\mathscr{H}$ is a function $B: \mathscr{H}\times \mathscr{H}\to \mathbb{C}$ satisfying
+
+1. $B(u,av+bw)=aB(u,v)+bB(u,w)$ for all $u,v,w\in \mathscr{H}$ and $a,b\in \mathbb{C}$. 
+2. $B(av+bw,u)=\overline{a}B(v,u)+\overline{b}B(w,u)$ for all $u,v,w\in \mathscr{H}$ and $a,b\in \mathbb{C}$.
+3. $|B(u,v)|\leq C\|u\|\|v\|$ for all $u,v\in \mathscr{H}$ and some constant $C>0$.
+
+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 $u,v\in \mathscr{H}$.
+
+[Proof ignored here]
+
+### The adjoint of a bounded operator
+
+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 $u,v\in \mathscr{H}$.
+
+[Proof ignored here]
+
+And $\|A^*\|=\|A\|$.
+
+Additional properties of bounded operators:
+
+Let $A,B\in \mathscr{B}(\mathscr{H})$ and $a,b\in \mathbb{C}$. Then
+
+1. $(aA+bB)^*=\overline{a}A^*+\overline{b}B^*$.
+2. $(AB)^*=B^*A^*$.
+3. $(A^*)^*=A$.
+4. $\|A^*\|=\|A\|$.
+5. $\|A^*A\|=\|A\|^2$.
+
+#### Definition of self-adjoint operator
+
+An operator $A\in \mathscr{B}(\mathscr{H})$ is self-adjoint if $A^*=A$.
+
+#### Definition of normal operator
+
+An operator $N\in \mathscr{B}(\mathscr{H})$ is normal if $NN^*=N^*N$.
+
+#### Definition of unitary operator
+
+An operator $U\in \mathscr{B}(\mathscr{H})$ is unitary if $U^*U=UU^*=I$.
+
+where $I$ is the identity operator on $\mathscr{H}$.
+
+#### Definition of orthogonal projection
+
+An operator $P\in \mathscr{B}(\mathscr{H})$ is an orthogonal projection if $P^*=P$ and $P^2=P$.
+
+### Tensor product of (infinite-dimensional) Hilbert spaces
+
+#### Definition of tensor product
+
+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 $\mathscr{H}_1\times \mathscr{H}_2$.
+
+$$
+(u_1\otimes u_2)(v_1,v_2)=\langle u_1,v_1\rangle_{\mathscr{H}_1}\langle u_2,v_2\rangle_{\mathscr{H}_2}
+$$
+
+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
+
+$$
+\langle u\otimes v,u'\otimes v'\rangle=\langle u,u'\rangle_{\mathscr{H}_1}\langle v,v'\rangle_{\mathscr{H}_2}
+$$
+
+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.
+
+Denoted by $\mathscr{H}_1\otimes \mathscr{H}_2$.
+
+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 $\mathscr{H}_2$.
+
+### Fock space
+
+#### Definition of Fock space
+
+Let $\mathscr{H}^{\otimes n}$ be the $n$-fold tensor product of $\mathscr{H}$.
+
+Set $\mathscr{H}^{\otimes 0}=\mathbb{C}$.
+
+The Fock space of $\mathscr{H}$ is the direct sum of all $\mathscr{H}^{\otimes n}$.
+
+$$
+\mathscr{F}(\mathscr{H})=\bigoplus_{n=0}^\infty \mathscr{H}^{\otimes n}
+$$
+
+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 $|\psi_0|^2+\sum_{n=1}^\infty \int|\psi_n(x_1,\cdots,x_n)|^2dx_1\cdots dx_n<\infty$.