I'm studying a part of Bott-Tu, and I stumbled upon this mystic sentence in the proof of the homotopy property of vector bundles (Theorem 6.8). The "near" part should be intended in the sense of a continuous transformation between linear maps, but does this mean that to be an isomorphism is a somehow stable property for linear maps? It's not something I've dealt with before...
Any help on this is really appreciated.
If $V,W$ are a pair of isomorphic vector spaces, the set $GL(V,W)$ of isomorphisms $V \to W$ is open in the space $\mathrm{Hom}(V,W)$ of linear maps $V \to W$, and thus any sufficiently small perturbation of an isomorphism is an isomorphism.
One way to prove this is to fix bases for $V,W$ so that $GL(V,W) \subset \mathrm{Hom}(V,W)$ becomes the set $GL(n,\mathbb R) \subset \mathbb R^{n\times n}$ of matrices (here $n$ is the common dimension of $V,W$), and note that $GL(n,\mathbb R)$ is the preimage under the continuous function $\det$ of the open set $\mathbb R \setminus \{0\}$.