# How to define orientation on infinite dimensional vector space

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?

• I don't think this will fly in general. For many of the $\Bbb{V}$ that you'd be likely to care about (e.g., $\ell^p$ for $1 \leq p < \infty$), $\operatorname{Aut}(\Bbb{V})$ is contractible (so, in particular, connected). See, e.g., here. – Micah Oct 6 '17 at 5:00
• There definitely are cases where infinite-dimensional objects are thought of as oriented (e.g., the loop space of a manifold $M$ is considered to be an oriented (Fréchet) manifold iff $M$ is spin) but I don't know enough to be able to say anything meaningful about the general theory (or even if there is one). – Micah Oct 6 '17 at 5:02
• This question came to my mind thinking about a definition of orientation for a Banach manifold. The basic one uses determinant, volume forms, or orientation on the tangent spaces. The first two don't work, so I tried to make sense of the latter in this case. – Grassy LittleRoot Oct 6 '17 at 5:14