L2 Space

analysis functional-analysis

Definition

$L^2[a,b]$

For $a,b\in \mathbb{R}$, the space $L^2[a,b]$ is the set of all complex-valued functions on $[a,b]$ whose square is Lebesgue integrable:

$$L^2[a,b]:=\left\{f:[a,b]\to \mathbb{C}|{\int }_a^b|f(x){|}^{2}dx<\infty \right\}.$$

Here the integral is understood as a Lebesgue integral. These functions are called square-integrable.

Functions that differ only on a set of measure zero are usually identified.

Hilbert Space

The space becomes a Hilbert space with inner product

$$⟨f,g⟩:={\int }_a^b\overline{f(x)}g(x)dx$$

where the integral is a Lebesgue integral. The induced norm is

$$\Vert f{\Vert }_2:={\left({\int }_a^b|f(x){|}^{2}dx\right)}^{1/2}.$$