- Let \(\mathcal { O } ( D )\) denote the space of holomorphic functions on the unit disk D and let
\[ { \mathcal H } = { \mathcal O } ( D ) \cap L ^ { 2 } ( D ) = \left\{ f \in { \mathcal O } ( D ) : \int _ { D } | f | ^ { 2 } d x d y < + \infty \right\} . \]
- Show that for all compact sets \(K \subset D\) , there is a constant \(C _ { K } \in \mathbb { R } ^ { + }\) such that
\[ \operatorname* { s u p } _ { z \in K } | f ( z ) | \leq C _ { K } \| f \| _ { L ^ { 2 } ( D ) } . \]
- Show that H is a closed subspace of \(L ^ { 2 } ( D )\) and hence is a Hilbert space.
Solution
Write \(dA=dx\,dy\). We identify each function in \(\mathcal H\) with its almost-everywhere class in \(L^2(D)\). This is injective: a continuous function nonzero at a point is bounded away from zero on a small disk of positive area, so cannot be zero almost everywhere.
1. A disk contained in \(D\) gives an \(L^2\) point-evaluation estimate.
For \(\overline{D(a,r)}\subset D\) and \(0<\rho<r\), Cauchy’s formula at the center and Cauchy–Schwarz give \[ |f(a)|^2\leq\frac1{2\pi}\int_0^{2\pi} |f(a+\rho e^{it})|^2\,dt \] [@SS03; @Fol13]. Multiply by \(2\pi\rho\) and integrate from zero to \(r\). Polar coordinates yield \[ \pi r^2|f(a)|^2\leq\int_{D(a,r)}|f(z)|^2\,dA(z). \] For nonempty compact \(K\subset D\), set \(r=(1-\max_{a\in K}|a|)/2>0\). The closed radius-\(r\) disk about every \(a\in K\) is contained in \(D\). Therefore \[ \sup_{a\in K}|f(a)|\leq C_K\|f\|_{L^2(D)}, \qquad C_K=\frac1{\sqrt\pi r}. \] For the empty compact set the bound has no points to check, and any positive \(C_K\) suffices. This proves (a).
2. An \(L^2\) limit of elements of \(\mathcal H\) has a holomorphic representative.
Suppose \(f_n\in\mathcal H\) and \(f_n\to F\) in \(L^2(D)\). Choose a measurable representative of \(F\). The sequence is Cauchy in \(L^2\), so step 1 applied to \(f_n-f_m\) makes it uniformly Cauchy on each compact subset of \(D\). Completeness of \(\mathbb C\) gives a pointwise limit \(f\), and the same estimates give uniform convergence on every compact set. The local uniform limit theorem makes \(f\) holomorphic on \(D\) [@SS03].
Pointwise convergence and Fatou’s lemma give \[ \int_D|f-F|^2\,dA \leq\liminf_{n\to\infty}\int_D|f_n-F|^2\,dA=0 \] [@Fol13]. Thus \(f=F\) almost everywhere, so \(f\) belongs to \(\mathcal H\) and represents the given \(L^2\) limit.
3. The space \(\mathcal H\) is a closed linear subspace and is complete.
Linear combinations of its functions are holomorphic and square integrable, so it is a linear subspace of \(L^2(D)\). Step 2 proves that this subspace is closed. The space \(L^2(D)\) is Hilbert [@Fol13]; a Cauchy sequence in a closed subspace converges in the ambient Hilbert space and has its limit in that subspace. Therefore \(\mathcal H\), with the inherited inner product, is a Hilbert space. This proves (b).