  In this work we study some algebraic and topological properties of the ring
${\mathcal O}(X^\nu)$ of global analytic functions of the normalization
$(X^\nu,{\mathcal O}_{X^\nu})$ of a reduced complex analytic space
$(X,{\mathcal O}_X)$. If $(X,{\mathcal O}_X)$ is a Stein space, we characterize
${\mathcal O}(X^\nu)$ in terms of the (topological) completion of the integral
closure $\overline{{\mathcal O}(X)}^\nu$ of the ring ${\mathcal O}(X)$ of
global holomorphic functions on $X$ (inside its total ring of fractions) with
respect to the usual Fr\'echet topology of $\overline{{\mathcal O}(X)}^\nu$.
This shows that not only the Stein space $(X,{\mathcal O}_X)$ but also its
normalization is completely determined by the ring ${\mathcal O}(X)$ of global
analytic functions on $X$. This result was already proved in 1988 by
Hayes-Pourcin when $(X,{\mathcal O}_X)$ is an irreducible Stein space whereas
in this paper we afford the general case. We also analyze the real underlying
structures $(X^{\mathbb R},{\mathcal O}_X^{\mathbb R})$ and $(X^{\nu\,{\mathbb
R}},{\mathcal O}_{X^\nu}^{\mathbb R})$ of a reduced complex analytic space
$(X,{\mathcal O}_X)$ and its normalization $(X^\nu,{\mathcal O}_{X^\nu})$. We
prove that the complexification of $(X^{\nu\,{\mathbb R}},{\mathcal
O}_{X^\nu}^{\mathbb R})$ provides the normalization of the complexification of
$(X^{\mathbb R},{\mathcal O}_X^{\mathbb R})$ if and only if $(X^{\mathbb
R},{\mathcal O}_X^{\mathbb R})$ is a coherent real analytic space. Roughly
speaking, coherence of the real underlying structure is equivalent to the
equality of the following two combined operations: (1) normalization + real
underlying structure + complexification, and (2) real underlying structure +
complexification + normalization.

Algebraic Geometry