Bargmann-Fock space ($\mathcal{F}=\{f\text{ entire}:{\pi }^{-1}{\int }_{\mathbb{C}}|f(z){|}^2{e}^{-|z{|}^2}dxdy<\infty \}$)

The Bargmann-Fock space is a Hilbert space of entire holomorphic functions with Gaussian-weighted area inner product. If $f(z)=\sum a_n{z}^n$, angular integration and the Gaussian integral give $\Vert f{\Vert }^2=\sum n!|a_n{|}^2$. Conversely any such square-summable coefficient sequence defines an entire function, since $\sum |a_n{z}^n|\le (\sum n!|a_n{|}^2{)}^{1/2}{e}^{|z{|}^2/2}$. This proves completeness and shows that $z^n/\sqrt{n!}$ is an orthonormal basis. The same bound shows boundedness of evaluation and gives reproducing kernel $e^{z\bar{w}}$.