Hilbert Space Theory

Inner Product Space

An inner product space is a vector space

V

over

𝔽

with a function

⟨ \cdot, \cdot ⟩ : V \times V \to 𝔽

that is positive definite, linear in one variable, and conjugate symmetric.

Hilbert Space

A Hilbert space is an inner product space that is complete with respect to the norm

‖ v ‖ = \sqrt{⟨ v, v ⟩}

Cauchy-Schwarz Inequality

Suppose that

V

is an inner product space and

u, v \in V

. Then

| ⟨ u, v ⟩ | \leq ‖ u ‖ ‖ v ‖ .

Parallelogram Identity

Suppose that

V

is an inner product space and

u, v \in V

. Then

{‖ u + v ‖}^{2} + {‖ u - v ‖}^{2} = 2 {‖ u ‖}^{2} + 2 {‖ v ‖}^{2} .

Orthogonal Complement

Suppose that

M \subseteq H

, where

H

is an inner product space. The orthogonal complement of

M

M^{⟂} : = {h \in H : ⟨ h, m ⟩ = 0 for every m \in M} .

Closest Point Projection onto Closed Convex Set

Suppose that

H

is a Hilbert space,

C \subseteq H

is nonempty, closed, and convex, and

h \in H

. Then there exists a unique

c \in C

such that

‖ h - c ‖ = \inf_{x \in C} ‖ h - x ‖ .

Orthogonal Decomposition

Suppose that

H

is a Hilbert space and

M

is a closed linear subspace of

H

. Then

H = M \oplus︎ M^{⟂} .

Orthonormal Family

A family

{(e_{j})}_{j \in J}

in an inner product space is orthonormal if

‖ e_{j} ‖ = 1

for every

j \in J

, and

⟨ e_{j}, e_{k} ⟩ = 0

whenever

j \neq k

Bessel's Inequality

Suppose that

{(e_{j})}_{j \in J}

is an orthonormal family in a Hilbert space

H

. Then for every

h \in H

\sum_{j \in J} {| ⟨ h, e_{j} ⟩ |}^{2} \leq {‖ h ‖}^{2} .

Parseval's Identity

Suppose that

{(e_{j})}_{j \in J}

is an orthonormal basis of a Hilbert space

H

. Then for every

h \in H

{‖ h ‖}^{2} = \sum_{j \in J} {| ⟨ h, e_{j} ⟩ |}^{2} .

Riesz Representation Theorem for Hilbert Spaces

Suppose that

H

is a Hilbert space and

φ : H \to 𝔽

is a bounded linear functional. Then there exists a unique

h \in H

such that

φ (x) = ⟨ x, h ⟩

for every

x \in H