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

• $$(1,0,0),(0,1,0),(0,0,1)$$

### $$L^2$$ 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

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