Let $\mathbb{V}$ be a real Banach space (if someone knows the answer for more arbitrary T.V.S. then great). Is there some concept of orientation that one could define on $\mathbb{V}$ that matches the already existing one for finite dimension?
My first guess was using the non-connectivity of $\mathrm{Aut}(\mathbb{V})$ analogously to the finite dimensional case.
Let $\mathrm{Aut}(\mathbb{V})$ be the group of bounded/continuous linear isomorphisms $\mathbb{V} \to \mathbb{V}$. If $\mathrm{Aut}(\mathbb{V})$ has to two connected components $C_{+}$ and $C_{-}$ such that $C_{+}$ is a subgroup and that $AB \in C_{+}$ for all $A,B \in C_{-}$, then I could take $C_{+}$ to be the "orientation preserving" maps and an orientation to be an orbit of the action of $C_{+}$ on the set of basis of $\mathbb{V}$.
In finite dimensions this condition is easy to check because of the determinant. Is this true for any Banach space? If not, is there a better way to generalize this concept?