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Zheyuan Wu
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pages/Math401/Math401_T2.md
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# Topic 2: Finite-dimensional Hilbert spaces
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Recall the complex number is a tuple of two real numbers, $z=(a,b)$ with addition and multiplication defined by
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$$
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(a,b)+(c,d)=(a+c,b+d)
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$$
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$$
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(a,b)\cdot(c,d)=(ac-bd,ad+bc)
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$$
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or in polar form,
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$$
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z=re^{i\theta}=r(\cos\theta+i\sin\theta)
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$$
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where $r=\sqrt{a^2+b^2}=\sqrt{z\overline{z}}$ and $\theta=\tan^{-1}(b/a)$.
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The complex conjugate of $z$ is $\overline{z}=(a,-b)$.
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## Section 1: Finite-dimensional Complex Vector Spaces
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Here, we use the field $\mathbb{C}$ of complex numbers. or the field $\mathbb{R}$ of real numbers as the field $\mathbb{F}$ we are going to encounter.
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### Definition of vector space
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A vector space $\mathscr{V}$ over a field $\mathbb{F}$ is a set equipped with an **addition** and a **scalar multiplication**, satisfying the following axioms:
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1. Addition is associative and commutative. For all $u,v,w\in \mathscr{V}$,
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Associativity:
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$$
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(u+v)+w=u+(v+w)
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$$
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Commutativity:
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$$
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u+v=v+u
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$$
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2. Additive identity: There exists an element $0\in \mathscr{V}$ such that $v+0=v$ for all $v\in \mathscr{V}$.
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3. Additive inverse: For each $v\in \mathscr{V}$, there exists an element $-v\in \mathscr{V}$ such that $v+(-v)=0$.
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4. Multiplicative identity: There exists an element $1\in \mathbb{F}$ such that $v\cdot 1=v$ for all $v\in \mathscr{V}$.
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5. Multiplicative inverse: For each $v\in \mathscr{V}$ and $c\in \mathbb{F}$, there exists an element $c^{-1}\in \mathbb{F}$ such that $v\cdot c^{-1}=1$.
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6. Distributivity: For all $u,v\in \mathscr{V}$ and $c,d\in \mathbb{F}$,
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$$
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c(u+v)=cu+cv
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$$
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A vector is an ordered pair of elements over the field $\mathbb{F}$.
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If we consider $\mathbb{F}=\mathbb{C}^n$, $n\in \mathbb{N}$, then $u=(a_1,a_2,\cdots,a_n), v=(b_1,b_2,\cdots,b_n)\in \mathbb{C}^n$ are vectors.
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The addition and scalar multiplication are defined by
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$$
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u+v=(a_1+b_1,a_2+b_2,\cdots,a_n+b_n)
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$$
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$$
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cu=(ca_1,ca_2,\cdots,ca_n)
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$$
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$c\in \mathbb{C}$.
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The matrix transpose is defined by
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$$
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u^T=(a_1,a_2,\cdots,a_n)^T=\begin{pmatrix}
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a_1 \\
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a_2 \\
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\vdots \\
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a_n
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\end{pmatrix}
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$$
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The complex conjugate transpose is defined by
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$$
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u^*=(a_1,a_2,\cdots,a_n)^*=\begin{pmatrix}
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\overline{a_1} \\
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\overline{a_2} \\
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\vdots \\
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\overline{a_n}
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\end{pmatrix}
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$$
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> In physics, the complex conjugate is sometimes denoted by $z^*$ instead of $\overline{z}$.
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> The complex conjugate transpose is sometimes denoted by $u^\dagger$ instead of $u^*$.
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### Hermitian inner product and norms
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On $\mathbb{C}^n$, the Hermitian inner product is defined by
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$$
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\langle u,v\rangle=\sum_{i=1}^n \overline{u_i}v_i
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$$
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The norm is defined by
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$$
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\|u\|=\sqrt{\langle u,u\rangle}
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$$
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#### Definition of Inner product
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Let $\mathscr{H}$ be a complex vector space. An inner product on $\mathscr{H}$ is a function $\langle \cdot, \cdot \rangle: \mathscr{H}\times \mathscr{H}\to \mathbb{C}$ satisfying the following axioms:
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1. For each $u\in \mathscr{H}$, $v\mapsto \langle u,v\rangle$ is a linear map.
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$$
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\langle u,av+bw\rangle=a\langle u,v\rangle+b\langle u,w\rangle
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$$
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For all $u,v,w\in \mathscr{H}$ and $a,b\in \mathbb{C}$.
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2. For all $u,v\in \mathscr{H}$, $\langle u,v\rangle=\overline{\langle v,u\rangle}$.
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$u\mapsto \langle u,v\rangle$ is a conjugate linear map.
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3. $\langle u,u\rangle\geq 0$ and $\langle u,u\rangle=0$ if and only if $u=0$.
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#### Definition of norm
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Let $\mathscr{H}$ be a complex vector space. A norm on $\mathscr{H}$ is a function $\|\cdot\|: \mathscr{H}\to \mathbb{R}$ satisfying the following axioms:
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1. For all $u\in \mathscr{H}$, $\|u\|\geq 0$ and $\|u\|=0$ if and only if $u=0$.
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2. For all $u\in \mathscr{H}$ and $c\in \mathbb{C}$, $\|cu\|=|c|\|u\|$.
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3. Triangle inequality: For all $u,v\in \mathscr{H}$, $\|u+v\|\leq \|u\|+\|v\|$.
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#### Definition of inner product space
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A complex vector space $\mathscr{H}$ with an inner product is called a **Hilbert space**.
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#### Cauchy-Schwarz inequality
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For all $u,v\in \mathscr{H}$,
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$$
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|\langle u,v\rangle|\leq \|u\|\|v\|
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$$
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#### Parallelogram law
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For all $u,v\in \mathscr{H}$,
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$$
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\|u+v\|^2+\|u-v\|^2=2(\|u\|^2+\|v\|^2)
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$$
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#### Polarization identity
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For all $u,v\in \mathscr{H}$,
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$$
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\langle u,v\rangle=\frac{1}{4}(\|u+v\|^2-\|u-v\|^2+i\|u+iv\|^2-i\|u-iv\|^2)
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$$
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#### Additional definitions
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Let $u,v\in \mathscr{H}$.
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$||v||$ is the length of $v$.
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$v$ is a unit vector if $\|v\|=1$.
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$u,v$ are orthogonal if $\langle u,v\rangle=0$.
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#### Definition of orthonormal basis
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A set of vectors $\{e_1,e_2,\cdots,e_n\}$ in a Hilbert space $\mathscr{H}$ is called an orthonormal basis if
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1. $\langle e_i,e_j\rangle=\delta_{ij}$ for all $i,j\in \{1,2,\cdots,n\}$.
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$$
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\delta_{ij}=\begin{cases}
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1 & \text{if } i=j \\
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0 & \text{if } i\neq j
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\end{cases}
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$$
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2. $n=\dim \mathscr{H}$.
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### Subspaces and orthonormal bases
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#### Definition of subspace
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A subset $\mathscr{W}$ of a vector space $\mathscr{V}$ is a subspace if it is closed under addition and scalar multiplication.
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#### Definition of orthogonal complement
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Let $E$ be a subset of a Hilbert space $\mathscr{H}$. The orthogonal complement of $E$ is the set of all vectors in $\mathscr{H}$ that are orthogonal to every vector in $E$.
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$$
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E^\perp=\{v\in \mathscr{H}: \langle v,w\rangle=0 \text{ for all } w\in E\}
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$$
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#### Definition of orthogonal projection
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Let $E$ be a $m$-dimensional subspace of a Hilbert space $\mathscr{H}$. An orthogonal projection of $E$ is a linear map $P_E: \mathscr{H}\to E$
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$$
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P_E(v)=\sum_{i=1}^m \langle v,e_i\rangle e_i
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$$
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#### Definition of orthonormal direct sum
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A inner product space $\mathscr{H}$ is the direct sum of $E_1,E_2,\cdots,E_n$ if
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$$
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\mathscr{H}=E_1\oplus E_2\oplus \cdots \oplus E_n
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$$
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and $E_i\cap E_j=\{0\}$ for all $i\neq j$.
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That is, $\forall v\in \mathscr{H}$, there exists a unique $v_i\in E_i$ such that $v=v_1+v_2+\cdots+v_n$.
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#### Definition of meet and join of subspaces
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Let $E$ and $F$ be two subspaces of a Hilbert space $\mathscr{H}$. The meet of $E$ and $F$ is the subspace $\mathscr{H}$ such that
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$$
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E\land F=E\cap F
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$$
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The join of $E$ and $F$ is the subspace $\mathscr{H}$ such that
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$$
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E\lor F=\{u+v: u\in E, v\in F\}
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$$
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### Null space and range
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#### Definition of null space
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Let $A$ be a linear map from a vector space $\mathscr{V}$ to a vector space $\mathscr{W}$. The null space of $A$ is the set of all vectors in $\mathscr{V}$ that are mapped to the zero vector in $\mathscr{W}$.
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$$
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\text{Null}(A)=\{v\in \mathscr{V}: Av=0\}
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$$
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#### Definition of range
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Let $A$ be a linear map from a vector space $\mathscr{V}$ to a vector space $\mathscr{W}$. The range of $A$ is the set of all vectors in $\mathscr{W}$ that are mapped from $\mathscr{V}$.
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$$
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\text{Range}(A)=\{w\in \mathscr{W}: \exists v\in \mathscr{V}, Av=w\}
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$$
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### Dual spaces and adjoints of linear maps\
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#### Definition of linear map
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A linear map $T: \mathscr{V}\to \mathscr{W}$ is a function that satisfies the following axioms:
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1. Additivity: For all $u,v\in \mathscr{V}$ and $a,b\in \mathbb{F}$,
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$$
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T(au+bv)=aT(u)+bT(v)
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$$
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2. Homogeneity: For all $u\in \mathscr{V}$ and $a\in \mathbb{F}$,
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$$
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T(au)=aT(u)
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$$
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#### Definition of linear functionals
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A linear functional $f: \mathscr{V}\to \mathbb{F}$ is a linear map from $\mathscr{V}$ to $\mathbb{F}$.
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Here, $\mathbb{F}$ is the field of complex numbers.
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#### Definition of dual space
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Let $\mathscr{V}$ be a vector space over a field $\mathbb{F}$. The dual space of $\mathscr{V}$ is the set of all linear functionals on $\mathscr{V}$.
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$$
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\mathscr{V}^*=\{f:\mathscr{V}\to \mathbb{F}: f\text{ is linear}\}
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$$
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If $\mathscr{H}$ is a finite-dimensional Hilbert space, then $\mathscr{H}^*$ is isomorphic to $\mathscr{H}$.
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Note $v\in \mathscr{H}\mapsto \langle v,\cdot\rangle\in \mathscr{H}^*$ is a conjugate linear isomorphism.
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#### Definition of adjoint of a linear map
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Let $T: \mathscr{V}\to \mathscr{W}$ be a linear map. The adjoint of $T$ is the linear map $T^*: \mathscr{W}\to \mathscr{V}$ such that
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$$
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\langle Tv,w\rangle=\langle v,T^*w\rangle
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$$
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for all $v\in \mathscr{V}$ and $w\in \mathscr{W}$.
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#### Definition of self-adjoint operators
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A linear operator $T: \mathscr{V}\to \mathscr{V}$ is self-adjoint if $T^*=T$.
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#### Definition of unitary operators
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A linear map $T: \mathscr{V}\to \mathscr{V}$ is unitary if $T^*T=TT^*=I$.
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### Dirac's bra-ket notation
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#### Definition of bra and ket
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Let $\mathscr{H}$ be a Hilbert space. The bra-ket notation is a notation for vectors in $\mathscr{H}$.
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$$
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\langle v|w\rangle
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$$
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is the inner product of $v$ and $w$.
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$$
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|v\rangle
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$$
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is the vector (or linear map) $v$.
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$$
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|u\rangle\langle v|
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$$
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is a linear map from $\mathscr{H}$ to $\mathscr{H}$.
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