0.1 Function spaces: notion and notations

A Hilbert space is a complete normed space whose norm is indexed by an inner (or scalar) product.

Two disjoint subspaces $A$ and $B$ of a space $S$ form a direct sum decomposition of $S$ if every element of $S$ can be written uniquely as a sum of an element of $A$ and an element of $B$ . The notation $S=A\oplus B$ is then used.

A measurable function $f$ belongs to the Lebesgue space $L_p\left(\mathbb{R}\right),1\le p<\infty$ if

An example of a Hilbert space is the Lebesgue space $L_2\left(\mathbb{R}\right)$ of measurable and square integrable functions. Indeed, the norm ${∥·∥}_2$ is induced by the scalar product

where $\overline{g\left(x\right)}$ denotes the complex conjugate of $g\left(x\right)$ . Two functions are said to be orthogonal in $L_2\left(\mathbb{R}\right)$ if their inner product is zero.

The Lebesgue measure can be replaced by a more general measure $\mu$ , leading to the weighted space ${\mathbf{L}}_2\left(\mu \right)$ , which has as inner product

and which contains the functions that have a finite norm ${∥f∥}_{d\mu }:={\sqrt{〈f,,,,f〉}}_{d\mu }<\infty$ .

A countable subset $\left\{f_k\right\}$ of functions belonging to a Hilbert space is a Riesz basis if every element $f$ of the space can be written uniquely as $f={\sum }_k{c}_k{f}_k$ , and if positive constants $A$ and $B$ exist such that

A Riesz basis is an orthogonal basis if the $f_k$ are mutually orthogonal. In this case, $A=B=1$ .