Let $X$ be a topological space, $H$ a separable Hilbert space and $E$ a vector bundle over $X$. Suppose $f \colon E \rightarrow X \times \mathbb{C}^n$ is a bundle isomorphism and $E$ is subbundle of a trivial bundle $X\times H$. Can $f$ be extended pointwise by $0$ (the extension $F$ is defined as $f$ on $E_x$ and $0$ on $(E_x)^\perp$) so that the extensions $F_x \colon H \rightarrow H$ form a family of linear maps that depends continuously on x?
Since $f$ is an isomorphism, $f_x \colon E_x \rightarrow \mathbb{C}$ is a continuous family of linear maps and since $E$ is a trivial vector bundle we can continuously choose a basis for $E_x$ to perform the extension, but I can not write a rigorous proof for it.