Hilbert space of square-integrable observables

For a measure space $(X,\Sigma,\mu)$ , the complex Hilbert space $L^2(X,\mu)$ consists of measurable $f:X\to\mathbb C$ with $\int |f|^2\,d\mu<\infty$ , identified modulo almost-everywhere equality, with

\langle f,h\rangle=\int_X f\overline h\,d\mu, \qquad \|f\|_2=\left(\int_X|f|^2\,d\mu\right)^{1/2}.

The inner product here is linear in its first argument. Completeness is the $p=2$ case of completeness of $L^p$ spaces.

Probability spaces

If $\mu(X)=1$ , then $1\in L^2$ , $\|1\|_2=1$ , and $L^\infty\subseteq L^2\subseteq L^1$ . The orthogonal complement of constants is the space of mean-zero observables.

Finite example

On a uniform $n$ -point space this is $\mathbb C^n$ with $\langle z,w\rangle=n^{-1}\sum_j z_j\overline{w_j}$ . As a Hilbert space it has dimension $n$ ; the separate function algebra on the same set uses coordinatewise multiplication.