If $X$ is a compact Hausdorff space, one may view the space of complex, continuous functions on it as the space of continuous sections in the trivial Hermitian bundle $X \times \mathbb C$. By the Riesz-Markov theorem, the dual of this space is the space of complex, regular Borel measures on $X$.

If $(E, h) \to X$ is a topological Hermitian vector bundle over $X$, of finite rank, and if $\Gamma (X, E)$ is the space of continuous sections in it, endowed with the norm $\| s \| = \sup _{x \in X} \sqrt {h_x \big( s(x), s(x) \big)}$, where $h_x$ is the Hermitian product in $E_x$ (the fiber over $x \in X$),

does the topological dual of $\Gamma (X, E)$ have a similarly nice description?

The bundle is assumed to be locally trivial, and if $X$ is paracompact, one may try to answer the question locally, and then glue the results with a partition of unity. The answer, though, is really ugly, and in particular seems to depend on the non-canonical choice of a partition of unity. I am looking for something deeper than this.

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    $\begingroup$ I am not sure at all (didn't think it through), but I think since there are too many local automorphisms of a continuous line bundle, the answer will be more-or-less tautological (to a subset of $X$ we associate a form on the sections of the dual bundle, stuff like this). To get more definite description, maybe one should use the Hermitian structure (consider balls etc.) - say if one considers real line bundle with inner product then there is only a slight twist by orientation or something like this (?). Maybe also one can consider a connection on the bundle to further "rigidify" the answer. $\endgroup$ – Sasha Feb 4 at 20:21

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