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 \(V× V → ℂ\)

Properties

Definition of norm

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

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