Normalization commutes with restriction to an open subvariety

First assume $X$ is irreducible. Cover $U$ by principal opens $D(f)$ contained in affine charts $V$ of $X$ [F2]. Over each such principal open the normalization restricts to the affine normalization with ring map $A_f\hookrightarrow B_f$, which is finite, surjective and birational and has normal source, because these properties hold for the affine normalization and are preserved by principal localization [F1]. The pieces agree on overlaps as subrings of the common function field $k(X)$ [F2], so ${\nu }^{-1}(U)\to U$ is finite (finiteness is affine-local on the target), surjective (each piece is), and induces the identity on function fields, hence is birational; and ${\nu }^{-1}(U)$ is normal because normality is local and each ${\nu }^{-1}(D(f))$ is normal [F1].

In the irreducible case these affine restrictions are exactly the defining integral-closure charts of the normalization of $U$, and their canonical overlap maps give ${\nu }^{-1}(U)\cong U^{\nu }$ over $U$. For reducible $X$, its normalization is the disjoint union of the component normalizations by [F1]; apply step 1.1 to each nonempty $U\cap X_i$ and omit components with empty intersection. These are the irreducible components of $U$, so their disjoint union is its normalization. Finiteness, surjectivity and normality hold componentwise, and birationality is read on each component.