Hilbert spaces

A Hilbert space is an abstract vector space possessing the structure of an inner product that allows length and angle to be measured. Also it is completed: there are enough limits in the space to allow the techniques of calculus to be used.

Definition

Inner product \(V× V → ℂ\)

• \(\langle x,y \rangle = \overline{\langle y,x \rangle}\)

• \(\langle a x_1 + b x_2, y \rangle = a \langle x_1, y \rangle + b \langle x_2,y \rangle\)

• \(\langle x,x \rangle \geq 0\)

Properties

• \(\langle x, a y_1 + b y_2 \rangle = \overline{a} \langle x, y_1 \rangle + \overline{b} \langle x,y_2 \rangle\)

Definition of norm

• \(|x| = \sqrt{\langle x,x \rangle)}\)

Orthonormal bases

Every Hilbert space admits an orthonormal basis, and each vector in the Hilbert space can be expanded as a series in terms of this orthonormal basis. An orthonormal basis in a Hilbert space satisfies

• \(\langle e_i, e_j \rangle = 0 \: when \: i \neq j\)
• \(|e_i| = 1\)
• \(e_i\)’s are dense in the Hilbert space (completeness)

x can be expressed as \[x = \sum_{i} a_i ei\]

with

\[a_i = \langle x, e_i \rangle\]

Examples

\(\mathbb{R^n}\) Euclidean space

Example of orthogonal basis

• \((1,0,0),(0,1,0),(0,0,1)\)

\(\mathbb{C^n}\) Euclidean space

\(l^2\) sequence space

\(L^2\) square integrable functions

• Definition of inner product \[ \langle f, g \rangle = \int_{-\infty}^{\infty} f(x) \overline{g(x)} dx\]

\(L^2(0,1)\) Fourier analysis

• Example of orthogonal basis

\[f_n(x) = e^{2 \pi i n x}\]

Spectral theory