In Example II.7.1.7 (iii) (p. 139) from Bruce Blackadar's "Operator Algebras – Theory of C*-Algebras and von Neumann Algebras" we have the following setup:
Let $V$ be a complex vector bundle of rank $n$ over a compact Hausdorff space $X$ with a Hermitian structure (i. e. a locally trivial bundle of $n$-dimensional Hilbert spaces over $X$) such that the family $\{\langle \cdot, \cdot \rangle_x \mid x \in X\}$ of inner products defined on each fiber varies continuously over $X$. Then the set $\Gamma(V)$ of continuous sections has a natural structure as a right $C(X, \mathbb{C})$-module and the pointwise inner product makes it into a pre-Hilbert-$C(X, \mathbb{C})$-module.
In the example it says that $\Gamma(V)$ is in fact a Hilbert-$C(X)$-module, i. e. that the norm induced by the inner product is complete.
It is obvious that if we have a Cauchy sequence $(\sigma_n)_n$ of continuous sections we end up with a rough section $\sigma$ defined as the pointwise limit of $(\sigma_n)_n$ on each fiber.
My question is: How do we prove that this is indeed a continuous section?
Best Answer
Let $V$ be your Hilbert bundle and $x \in X$. By local triviality, pick an open neighborhood $W$ of $x$ such that the bundle is trivial, so we are working with bounded continuous functions $W \to H$. We want to show that for every $\epsilon$ there is an open subset $U$ containing $x$ such that $\| \sigma(x) - \sigma(y) \|_{H} \leq \epsilon$ for every $y \in U$. But we have $$ \sigma(x) - \sigma(y) = \big(\sigma(x) - \sigma_n(x)\big) + \big(\sigma_n(x) - \sigma_n(y)\big) +\big(\sigma_n(y) - \sigma(y)\big). $$ Now, apply an $\epsilon/3$-argument. Pick $n$ large enough so that the first and last terms are $\leq \epsilon/3$ in the $H$-norm, which you can do by uniformity. Then, choose $U$ so that $\| \sigma_n(x) - \sigma_n(y) \|_{H} \leq \epsilon / 3$, which you can do by continuity.