I'm reading Karoubi's $K$-Theory, and am a bit uncertain of what he's calling a natural topology:
Let $X$ be a topological space, and let $V,V'$ be vector spaces over a field $k$. Let $X\times V,X\times V'$ be trivial vector bundles, and let $\mathscr{L}\big(V,V'\big)$ denote the space of linear functions $g:V\rightarrow V'$.
Karoubi gives a theorem that a certain map $\overline{g}:X\rightarrow \mathscr{L}\big(V,V'\big)$ is continuous with respect to the natural topology on $\mathscr{L}\big(V,V'\big)$.
Does he mean the compact-open topology? If so, it was never required that $V$ be a topological vector space, or even that $k$ be a topological field. I'm aware that the compact-open topology can be fashioned so long as there is a topology on $k$, but this was never required. Is he making these assumptions implicitly, or is there an alternative topology that he is referring to?
As Karoubi states at the very start of the first chapter, $k$ is restricted to be either $\mathbb{R}$ or $\mathbb{C}$. Note that the natural topology on $\mathscr{L}(V,V')$ is just its unique topological vector space topology (the unique topology that makes any linear isomorphism $k^N\to\mathscr{L}(V,V')$ a homeomorphism; note that in this context $V$ and $V'$ are assumed to be finite-dimensional). This turns out to coincide with the compact-open topology but that is a nontrivial fact and usually the topology is not thought of as the compact-open topology.