From 7d85e7ce5e3406d50e083654ee1d6c863fb5e4fe Mon Sep 17 00:00:00 2001 From: Zheyuan Wu <60459821+Trance-0@users.noreply.github.com> Date: Wed, 18 Jun 2025 15:39:13 -0500 Subject: [PATCH] update --- pages/Math401/Math401_T3.md | 176 ++++++++++++++++++ .../Math401/L2_square_integrable_problem.png | Bin 0 -> 6903 bytes 2 files changed, 176 insertions(+) create mode 100644 public/Math401/L2_square_integrable_problem.png diff --git a/pages/Math401/Math401_T3.md b/pages/Math401/Math401_T3.md index 5d7ef65..63c061c 100644 --- a/pages/Math401/Math401_T3.md +++ b/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 $|\psi_0|^2+\sum_{n=1}^\infty \int|\psi_n(x_1,\cdots,x_n)|^2dx_1\cdots dx_n<\infty$. diff --git a/public/Math401/L2_square_integrable_problem.png b/public/Math401/L2_square_integrable_problem.png new file mode 100644 index 0000000000000000000000000000000000000000..b2f9d5a4493c14165793daf4e1d65c5063cd6aa5 GIT binary patch literal 6903 zcmaKR1yIyq)b=Wp64Ko&B?w4&iy$eTQj&tOlv=t&Vx?i} z`tIs{Gw;kh^D@BB{_ef!o_o%@&pGG$MQdw5CL*9CfIuKbPgIq3!RICTtH#3t{}t@% z7Qn|XsKOIHJn$2MXB!2+(|Rf!c|LP{;pt=LVF$5yb$e;Y3$^jEvvY+yxOwhiwaS1= zyqHOf9(Goqj&80jdX6vcAcl6fECOOIYF@4^g8YKQEc}lo1Vtr;g;=yTSQM4?TsXt0 zAP_q3Cra{qz8PDyetLRqr?|Th`}hSPEYYRNzqU$J#umRxpb{nE)$8aXW4OxoK0{9 z9Ueva@6kgKld`emo}@x%;pUL9-@Yx56hAAfsId18@bVJwM@aXrPq06Z&(6>9O4|SN zEcBxVi>CzRSea>LON-3rL{-Q3p3^rvToR_v_BXd_1FuReT4DmvkJ_GR?|8sHN6n2zFvteYBnkFDll!VMKRw;&gRiNt zrx)JYsT3X_&LSo@vAvhdXRhGl@}M-U>eZqGgG@l0bM$Yn#a})*0w$aNMajs?t89D8 zTGZVqaug$9U7WaoDlBA|m1T~l;{D;vyS~T3A_Gc{~564#D~Otf4_-bGnv9_Ow?k+o1H6ug9v?piDta zi$vt*lx^+ejqI;qDUyTTl$8`$L=wY9YsJj&3?Q?Y6f$1g}v zLm&`hK^J08&COMg$ooxu-SmSFLcV)l)VP-!F);+TwzhSsTDEewk^*Y7u(lApzLcDz zB0N@B)-V-I@I5}B_!bUM?zeB0i;Iie+{rY;PVZ4^ha2O|NG(}mX@r72fsuG z+?P77DtcEukge?O?2Bt_qMK>SJcfet^Mp8ID38~1E}<@G%RLF{FEU7{3pCQZ*XvNo z!p1tXCBD!LkN&=gtx}vR?BN#)ERSIdQC=AAh-1;(`M^&@2)y!IvoOq{^o>fMyqp}Q zp`l@6c^S*i&CS!x%L=_dRG^Oi;c3Ltk@v~zY1?!D?BZg4;9Wcdf~C`1QUrJ&9uXmk z(2`3de~EM!qT8Hpq&`^vrD$cvdUaS4jKLzK8W-9WUek!*zimI8d?QfHCu{WCQStYe zJldX(jf*3^I2{O9F*T*H2|i(E3Ovpn-6%?;<0b3=lJkU@GN#s^zIede;&YsRE5mf-nTt zvM08@1rjXJ5QpO6gxx6A%2Khj%U2JI2VoydjqvHUkDYy++L{n?mLz??=Xmc&(%kqs z^|M0FE?{KH^TBbgOq7Kq?H@Y2J%qZ9`3y1RDPhvP_ESE^#9Nc-#l`JY8^@Q<9Uq0A zXZNc+Uu^vu74zM_KEVw8!FGr$|+bYYC^nNu8XZ%I9-e2lHFK`H09P&=+T8zrV%$d_?qI&UvtP+SQ&F;nvDt69luuD{S=XoD#Wa4ao0#>! 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