Past exam of the mathematics course of the University of Cambridge / 2010 / iii / Paper 7 / 4 / b

Every entire holomorphic function has a Taylor series $f(z)={\sum }_{n\ge 0}{a}_n{z}^n$, uniformly convergent on each circle. Angular orthogonality givesIntegrate in polar coordinates and use Tonelli theorem for the nonnegative sum:The last integral equals $n!/2$ by substituting $u=r^2$ and repeated integration by parts.

Conversely, for any $(b_n)\in {\ell }^2$, set $a_n=b_n/\sqrt{n!}$. On $|z|\le R$, the Cauchy-Schwarz inequality givesApplied to the tails, the same bound proves uniform convergence on every disk. The series therefore defines an entire holomorphic function, and the norm identity puts it in the Bargmann-Fock space $\mathcal{F}$. The coefficient map $f\mapsto (\sqrt{n!}{a}_n)$ is consequently an onto linear isometry $\mathcal{F}\to {\ell }^2$; polarization also identifies the inner products. Completeness of ${\ell }^2$ proves that $\mathcal{F}$ is a Hilbert space. Its standard coordinate orthonormal basis corresponds toThe norm identity shows both their unit norms and completeness; it also gives $|f(z)|\le e^{|z{|}^2/2}\Vert f{\Vert }_{\mathcal{F}}$.