[proofplan] The construction is local. A reduced analytic germ is represented, after a generic projection, as a finite branched cover of a polydisc; the integral closure of its local analytic algebra is finite by the analytic Noether normalization and Weierstrass preparation theorem. These finite local normal algebras glue because integral closure is intrinsic in the total quotient ring, producing a finite holomorphic map with the required universal property. [/proofplan] [step:Construct the local normal algebra] Let $(V,p)$ be a reduced analytic germ. By local Noether normalization for analytic algebras, after choosing suitable coordinates there is a finite map from $(V,p)$ to a polydisc germ $(\Delta^d,0)$, where $d=\dim_p V$. The local ring $\mathcal{O}_{V,p}$ is therefore a finite module over the regular local ring $\mathcal{O}_{\mathbb{C}^d,0}$. The Weierstrass preparation theorem implies that the integral closure of $\mathcal{O}_{V,p}$ in its total ring of fractions is again a finite $\mathcal{O}_{\mathbb{C}^d,0}$-module, hence a finite $\mathcal{O}_{V,p}$-module. [/step] [step:Realize the integral closure as an analytic space] A finite $\mathcal{O}_{V,p}$-algebra is represented locally by a coherent sheaf of algebras. Taking the analytic spectrum of this finite algebra gives a germ $(\widetilde V,\tilde p)$ and a finite holomorphic map $\nu:(\widetilde V,\tilde p)\to(V,p)$. Its local ring is integrally closed by construction, and the inclusion $\mathcal{O}_{V,p}\hookrightarrow\widetilde{\mathcal{O}}_{V,p}$ is exactly the integral closure inclusion. [/step] [step:Glue the local constructions] On overlaps of two local coordinate constructions, both algebras are the integral closure of the same reduced local analytic algebra inside the same total quotient sheaf. Integral closure is uniquely characterized by this property, so the local finite algebras are canonically isomorphic on overlaps. The cocycle condition is automatic from uniqueness. Therefore the local analytic spaces glue to a global normal analytic space $\widetilde V$ and a finite holomorphic map $\nu:\widetilde V\to V$. [/step] [step:Verify uniqueness and coherence] If $W\to V$ is another normalization, then at every point its local ring is the integral closure of $\mathcal{O}_{V,p}$ in the total quotient ring. Hence it is canonically isomorphic to the local ring of $\widetilde V$ over $p$. These local isomorphisms glue, giving uniqueness up to unique isomorphism. Since the normalization map is finite, the structure sheaf is a finite coherent algebra over $\mathcal{O}_V$; coherence of $\widetilde{\mathcal{O}}_V$ follows from finiteness and the coherence theorem for analytic sheaves. [/step]