I need to show any of the following results:
Consider $K=\mathbb{R}$ or $\mathbb{C}$, then,
1) The compact-open topology and the usual topology of $GL_n(K)$ are the same.
2) Taking inverses and multiplication are continuous in $GL_n(K)$ with the compact-open topology.
or
3) Give the bijection between $(X\times K^n)^{K^n}$ and $X^{GL_n(K)}$ and show that $\alpha\in (X\times K^n)^{K^n}$ is continuous iff $\hat{\alpha}\in X^{GL_n(K)}$ is continuous, where $\hat{\alpha}$ is its associated function.
I need any of these results in order to prove that a morphism of vector bundles is an isomorphism when it is a linear isomorphism in each fiber. I think proving 1) would be the best, but at this point any of the other results will do.
I'm completely stuck with it, so I would really appreciate any kind of help that can get me on track.
Since $K^n$ is a metric space, the compact-open topology coincides with the topology of uniform convergence on compact subsets of the domain; also, since the topology of $K^n$ has a countable basis and it is $\sigma$-compact, the compact-open topology itself has a countable basis, and it is therefore determined by its convergent sequences. You can check at once that a sequence of linear maps converges uniformly on all compact sets iff it converges uniformly on the unit ball, and more or less by definition a sequence of linear maps converges uniformly on the unit ball iff it converges in the topology given by norm of $\hom(K^n,K^n)$. This last topology is the «usual» topology you refer to.