Let $(X, \mathfrak{B}, \mu)$ be a measure space. Let $L^2(X)$ denote the $L^2$ -space associated with this measure space, i.e. $L^2(X)$ consists of measurable functions $f:X \longrightarrow \mathbb{C}$ such that

It is known that all $L^p$ -spaces, with $1\leq p \leq \infty$ , are Banach spaces with respect to the $L^p$ -norm $\;\|\cdot\|_p$ . For $L^2(X)$ we can say even more:

Theorem - $L^2(X)$ is an Hilbert Space with respect to the inner product $\langle \cdot, \cdot \rangle$ defined by

Proof:

Sequilinearity follows from the linearity of the Lebesgue integral (that is, the inner product defined above is linear in the second argument and conjugate linear in the first one). The conjugate symmetry is evident.

Positive definiteness holds by construction: If $\int_X |f|^2 d\mu = 0$ , then $|f|^2$ (and therefore $f$ ) is zero almost everywhere, thus the equivalence class of $f$ is the equivalence class of the zero function (which is the additive neutral element of the space).

Completeness is proved for the general case of $L^p$ -spaces in this article.$\square$

Remarks

• The spaces $\mathbb{C}^n$ or $\mathbb{R}^n$ with the usual inner product are particular examples of $L^2(X)$ , choosing $X = \{1, \dots, n\}$ with the counting measure.
• Choosing appropriate spaces $X$ it can be shown that all Hilbert spaces are isometrically isomorphic to a $L^2$ -space.