Bargmann-Fock space
= Bargmann-Fock space
{c}
{title2=$\mathcal F=\{f\text{ entire}:\pi^{-1}\int_{\mathbb C}|f(z)|^2e^{-|z|^2}\,dx\,dy<\infty\}$}
The Bargmann-Fock space is a <Hilbert space> of entire <holomorphic functions> with Gaussian-weighted area inner product. If $f(z)=\sum a_nz^n$, angular integration and the <Gaussian integral> give $\|f\|^2=\sum n!|a_n|^2$. Conversely any such square-summable coefficient sequence defines an entire function, since $\sum|a_nz^n|\leq(\sum n!|a_n|^2)^{1/2}e^{|z|^2/2}$. 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 $e^{z\bar w}$.