$A^2(\mathrm{\Omega })$ is closed, and the Bergman kernel is the sum over any complete orthonormal system

The closedness assertion is the closed-subspace conclusion already proved in The Bergman space $A^2(\mathrm{\Omega })$ and the Bergman kernel; the inner product and pointwise representatives are those fixed there.

Fix $w\in \mathrm{\Omega }$ and let $P_F{k}_w:={\sum }_{j\in F}⟨k_w,e_j⟩e_j$ for finite $F\subseteq J$. By [F3], $P_F{k}_w\to k_w$ in $A^2(\mathrm{\Omega })$. For each $j$, the reproducing identity in [F15] and conjugate symmetry [F12] give $⟨k_w,e_j⟩=\overline{e_j(w)}$, so $P_F{k}_w(z)={\sum }_{j\in F}{e}_j(z)\overline{e_j(w)}$. For every nonempty compact $K\subseteq \mathrm{\Omega }$, the compact evaluation bound [F2] gives ${\sup }_{z\in K}\mathrm{\mid }{P}_F{k}_w(z)-K_{\mathrm{\Omega }}(z,w)\mathrm{\mid }\le C_K\mathrm{\parallel }{P}_F{k}_w-k_w{\mathrm{\parallel }}_{L^2(\mathrm{\Omega })}\to 0$; the empty case is vacuous.

Fix $w_0\in \mathrm{\Omega }$. By [F9] choose $r>0$ with $B(w_0,r)\subseteq \mathrm{\Omega }$, and put $Q=\bar{B}(w_0,r\mathrm{/}2)$, a compact subset of $\mathrm{\Omega }$. For $w,w^{\mathrm{\prime }}\in B(w_0,r\mathrm{/}4)$ the segment $\gamma (t)=w+t(w^{\mathrm{\prime }}-w)$, $0\le t\le 1$, stays in $Q$ by the triangle inequality. Write ${ϵ}_{\mathrm{\ell }}$ for the multi-index with a $1$ in coordinate $\mathrm{\ell }$ and zeros elsewhere. For $f\in A^2(\mathrm{\Omega })$ with $\mathrm{\parallel }f{\mathrm{\parallel }}_2\le 1$, [F7] gives $\frac{d}{dt}f(\gamma (t))={\sum }_{\mathrm{\ell }<m}{\mathrm{\partial }}_{z_{\mathrm{\ell }}}f(\gamma (t))(w_{\mathrm{\ell }}^{\mathrm{\prime }}-w_{\mathrm{\ell }})$; by [F2] and finite Cauchy–Schwarz [F5], its modulus is at most $M_Q\mathrm{\parallel }{w}^{\mathrm{\prime }}-w\mathrm{\parallel }$, where $M_Q=({\sum }_{\mathrm{\ell }<m}{C}_{Q,{ϵ}_{\mathrm{\ell }}}^2{)}^{1\mathrm{/}2}$. The curve is continuous by [F6], so [F8] yields $\mathrm{\mid }f(w^{\mathrm{\prime }})-f(w)\mathrm{\mid }\le M_Q\mathrm{\parallel }{w}^{\mathrm{\prime }}-w\mathrm{\parallel }$. Taking the supremum over the unit ball of $A^2$ proves $\mathrm{\parallel }{E}_{w^{\mathrm{\prime }}}-E_w\mathrm{\parallel }\le M_Q\mathrm{\parallel }{w}^{\mathrm{\prime }}-w\mathrm{\parallel }$; by the isometry in [F12], $\mathrm{\parallel }{k}_{w^{\mathrm{\prime }}}-k_w{\mathrm{\parallel }}_2\le M_Q\mathrm{\parallel }{w}^{\mathrm{\prime }}-w\mathrm{\parallel }$. Thus $w\mapsto k_w$ is locally Lipschitz and continuous.

If $C$ is a compact subset of $A^2(\mathrm{\Omega })$ in its norm metric, the net $(P_{F}x)$ converges to $x$ uniformly for $x\in C$; the empty case is vacuous. For $\epsilon >0$, the norm balls of radius $\epsilon \mathrm{/}3$ centered at points of $C$ are open by the triangle inequality and form a cover in that metric, so [F13] gives a finite subcover with centers $x_1,\dots ,x_N$. For each center choose a finite $F_i$ with $\mathrm{\parallel }{P}_F{x}_i-x_i\mathrm{\parallel }<\epsilon \mathrm{/}3$ whenever $F\supseteq F_i$, using [F3]. For $F_0={\bigcup }_i{F}_i$ and $F\supseteq F_0$, contraction of $I-P_F$ from [F4] gives ${\sup }_{x\in C}\mathrm{\parallel }{P}_{F}x-x\mathrm{\parallel }<2\epsilon \mathrm{/}3$. By [F9] and step 1.3, $k(K)$ and $k(L)$ are compact for compact $K,L\subseteq \mathrm{\Omega }$, so this uniform convergence applies to both section families.

Fix $w\in \mathrm{\Omega }$. By [F16], the support of $(⟨k_w,e_j⟩{)}_{j\in J}$ is at most countable; if it is finite the expansion is finite, and otherwise [F10] enumerates it by a sequence. The corresponding finite partial sums converge to $k_w$ in norm by [F3] and uniformly on every compact subset in the variable $z$ by [F2]. Each partial sum is holomorphic in $z$ by [F6], so [F11] makes $z\mapsto K_{\mathrm{\Omega }}(z,w)$ holomorphic.

Let $S_F(z,w):={\sum }_{j\in F}{e}_j(z)\overline{e_j(w)}=P_F{k}_w(z)$ as in step 1.2. Step 2.1 and [F2] show $S_F\to K_{\mathrm{\Omega }}$ uniformly on each $K\times L$ with $K,L\subseteq \mathrm{\Omega }$ compact. For $\epsilon >0$, apply step 2.1 to the compact sets $k(K)$ and $k(L)$ with tolerance $\sqrt{\epsilon }$, and take the union $F_0$ of the resulting finite index sets. For every finite $G\subseteq J\setminus F_0$, finite Cauchy–Schwarz [F5] and the orthogonal projections [F4], with $F=F_0$, give ${\sum }_{j\in G}\mathrm{\mid }{e}_j(z)\overline{e_j(w)}\mathrm{\mid }\le \mathrm{\parallel }{P}_G{k}_z{\mathrm{\parallel }}_2\mathrm{\parallel }{P}_G{k}_w{\mathrm{\parallel }}_2\le \mathrm{\parallel }{k}_z-P_{F_0}{k}_z{\mathrm{\parallel }}_2\mathrm{\parallel }{k}_w-P_{F_0}{k}_w{\mathrm{\parallel }}_2<\epsilon$ uniformly on $K\times L$. Thus the series converges absolutely and uniformly there. Any compact subset of $\mathrm{\Omega }\times \mathrm{\Omega }$ is contained in the product of its compact coordinate projections by [F14], so the convergence holds on every such compact subset.

For $z,w\in \mathrm{\Omega }$, $K_{\mathrm{\Omega }}(z,w)=⟨k_w,k_z⟩$ by the reproducing identity [F15], so conjugate symmetry [F12] gives $K_{\mathrm{\Omega }}(w,z)=\overline{K_{\mathrm{\Omega }}(z,w)}$. Thus the kernel is antiholomorphic in $w$ as well as holomorphic in $z$ by step 2.2. Since step 3.1 identifies every complete orthonormal system's sum with the kernel defined in [F15], the expansion is basis-independent.