NoteNextra-origin/pages/Math401/Math401_T2.md

Topic 2: Finite-dimensional Hilbert spaces

Recall the complex number is a tuple of two real numbers, z=(a,b) with addition and multiplication defined by

(a,b)+(c,d)=(a+c,b+d)

(a,b)\cdot(c,d)=(ac-bd,ad+bc)

or in polar form,

z=re^{i\theta}=r(\cos\theta+i\sin\theta)

where r=\sqrt{a^2+b^2}=\sqrt{z\overline{z}} and \theta=\tan^{-1}(b/a).

The complex conjugate of z is \overline{z}=(a,-b).

Section 1: Finite-dimensional Complex Vector Spaces

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.

Definition of vector space

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:

1. Addition is associative and commutative. For all u,v,w\in \mathscr{V},

Associativity:

(u+v)+w=u+(v+w)

Commutativity:

u+v=v+u

2. Additive identity: There exists an element 0\in \mathscr{V} such that v+0=v for all v\in \mathscr{V}.

3. Additive inverse: For each v\in \mathscr{V}, there exists an element -v\in \mathscr{V} such that v+(-v)=0.

4. Multiplicative identity: There exists an element 1\in \mathbb{F} such that v\cdot 1=v for all v\in \mathscr{V}.

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.

6. Distributivity: For all u,v\in \mathscr{V} and c,d\in \mathbb{F},

c(u+v)=cu+cv

A vector is an ordered pair of elements over the field \mathbb{F}.

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.

The addition and scalar multiplication are defined by

u+v=(a_1+b_1,a_2+b_2,\cdots,a_n+b_n)

cu=(ca_1,ca_2,\cdots,ca_n)

c\in \mathbb{C}.

The matrix transpose is defined by

u^T=(a_1,a_2,\cdots,a_n)^T=\begin{pmatrix}
a_1 \\
a_2 \\
\vdots \\
a_n
\end{pmatrix}

The complex conjugate transpose is defined by

u^*=(a_1,a_2,\cdots,a_n)^*=\begin{pmatrix}
\overline{a_1} \\
\overline{a_2} \\
\vdots \\
\overline{a_n}
\end{pmatrix}

In physics, the complex conjugate is sometimes denoted by z^* instead of \overline{z}. The complex conjugate transpose is sometimes denoted by u^\dagger instead of u^*.

Hermitian inner product and norms

On \mathbb{C}^n, the Hermitian inner product is defined by

\langle u,v\rangle=\sum_{i=1}^n \overline{u_i}v_i

The norm is defined by

\|u\|=\sqrt{\langle u,u\rangle}

Definition of Inner product

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:

1. For each u\in \mathscr{H}, v\mapsto \langle u,v\rangle is a linear map.

\langle u,av+bw\rangle=a\langle u,v\rangle+b\langle u,w\rangle

For all u,v,w\in \mathscr{H} and a,b\in \mathbb{C}.

2. For all u,v\in \mathscr{H}, \langle u,v\rangle=\overline{\langle v,u\rangle}.

u\mapsto \langle u,v\rangle is a conjugate linear map.

3. \langle u,u\rangle\geq 0 and \langle u,u\rangle=0 if and only if u=0.

Definition of norm

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:

1. For all u\in \mathscr{H}, \|u\|\geq 0 and \|u\|=0 if and only if u=0.

2. For all u\in \mathscr{H} and c\in \mathbb{C}, \|cu\|=|c|\|u\|.

3. Triangle inequality: For all u,v\in \mathscr{H}, \|u+v\|\leq \|u\|+\|v\|.

Definition of inner product space

A complex vector space \mathscr{H} with an inner product is called a Hilbert space.

Cauchy-Schwarz inequality

For all u,v\in \mathscr{H},

|\langle u,v\rangle|\leq \|u\|\|v\|

Parallelogram law

For all u,v\in \mathscr{H},

\|u+v\|^2+\|u-v\|^2=2(\|u\|^2+\|v\|^2)

Polarization identity

For all u,v\in \mathscr{H},

\langle u,v\rangle=\frac{1}{4}(\|u+v\|^2-\|u-v\|^2+i\|u+iv\|^2-i\|u-iv\|^2)

Additional definitions

Let u,v\in \mathscr{H}.

||v|| is the length of v.

v is a unit vector if \|v\|=1.

u,v are orthogonal if \langle u,v\rangle=0.

Definition of orthonormal basis

A set of vectors \{e_1,e_2,\cdots,e_n\} in a Hilbert space \mathscr{H} is called an orthonormal basis if

1. \langle e_i,e_j\rangle=\delta_{ij} for all i,j\in \{1,2,\cdots,n\}.

\delta_{ij}=\begin{cases}
1 & \text{if } i=j \\
0 & \text{if } i\neq j
\end{cases}

2. n=\dim \mathscr{H}.

Subspaces and orthonormal bases

Definition of subspace

A subset \mathscr{W} of a vector space \mathscr{V} is a subspace if it is closed under addition and scalar multiplication.

Definition of orthogonal complement

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.

E^\perp=\{v\in \mathscr{H}: \langle v,w\rangle=0 \text{ for all } w\in E\}

Definition of orthogonal projection

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

P_E(v)=\sum_{i=1}^m \langle v,e_i\rangle e_i

Definition of orthonormal direct sum

A inner product space \mathscr{H} is the direct sum of E_1,E_2,\cdots,E_n if

\mathscr{H}=E_1\oplus E_2\oplus \cdots \oplus E_n

and E_i\cap E_j=\{0\} for all i\neq j.

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.

Definition of meet and join of subspaces

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

E\land F=E\cap F

The join of E and F is the subspace \mathscr{H} such that

E\lor F=\{u+v: u\in E, v\in F\}

Null space and range

Definition of null space

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}.

\text{Null}(A)=\{v\in \mathscr{V}: Av=0\}

Definition of range

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}.

\text{Range}(A)=\{w\in \mathscr{W}: \exists v\in \mathscr{V}, Av=w\}

Dual spaces and adjoints of linear maps\

Definition of linear map

A linear map T: \mathscr{V}\to \mathscr{W} is a function that satisfies the following axioms:

1. Additivity: For all u,v\in \mathscr{V} and a,b\in \mathbb{F},

T(au+bv)=aT(u)+bT(v)

2. Homogeneity: For all u\in \mathscr{V} and a\in \mathbb{F},

T(au)=aT(u)

Definition of linear functionals

A linear functional f: \mathscr{V}\to \mathbb{F} is a linear map from \mathscr{V} to \mathbb{F}.

Here, \mathbb{F} is the field of complex numbers.

Definition of dual space

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}.

\mathscr{V}^*=\{f:\mathscr{V}\to \mathbb{F}: f\text{ is linear}\}

If \mathscr{H} is a finite-dimensional Hilbert space, then \mathscr{H}^* is isomorphic to \mathscr{H}.

Note v\in \mathscr{H}\mapsto \langle v,\cdot\rangle\in \mathscr{H}^* is a conjugate linear isomorphism.

Definition of adjoint of a linear map

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

\langle Tv,w\rangle=\langle v,T^*w\rangle

for all v\in \mathscr{V} and w\in \mathscr{W}.

Definition of self-adjoint operators

A linear operator T: \mathscr{V}\to \mathscr{V} is self-adjoint if T^*=T.

Definition of unitary operators

A linear map T: \mathscr{V}\to \mathscr{V} is unitary if T^*T=TT^*=I.

Dirac's bra-ket notation

Definition of bra and ket

Let \mathscr{H} be a Hilbert space. The bra-ket notation is a notation for vectors in \mathscr{H}.

\langle v|w\rangle

is the inner product of v and w.

|v\rangle

is the vector (or linear map) v.

|u\rangle\langle v|

is a linear map from \mathscr{H} to \mathscr{H}.