Tensor product vector bundle

Stream: Is there code for X?

[Michael Rothgang]

Ideally, as in: the tensor product bundle of two smooth vector bundles is smooth.

I know we have the Hom bundle, direct sum and product bundles (though the latter seems slightly more awkward to work with).

[Sébastien Gouëzel]

There is a serious difficulty here that there is no canonical topology on the tensor product of two (topological, or even normed) vector spaces -- more precisely, there are several natural candidates, which don't coincide in general. Because of this, no-one has defined the tensor product of continuous vector bundles yet.

[Eric Wieser]

Is there a sense in which they're "canonical enough", in the way that we put the not-necessarily-canonical sup-norm on pi types?

[Anatole Dedecker]

The thing is the space don’t even coincide here (they do in finite dimensions of course), because you take the completions for different norms. Of course you could do something in finite dimensions, but the point is that there are nontrivial decisions to be taken

[Sébastien Gouëzel]

Here are the two main norms on tensor products. The space of finite rank operators from E to F has the canonical operator norm, and it is canonically identified with $F \otimes E^*$, so this gives one canonical norm on $F \otimes E^*$. Also, the space of bilinear forms on $E \times F$ is canonically identified with $E^* \otimes F^*$, and the space of bilinear forms has a canonical norm, so this gives a canonical norm on $E^* \otimes F^*$. The two norms I've just described are in general not equivalent (except in finite dimension), and the completion of the spaces for these two norms are therefore genuinely different. And one can not argue that one is more canonical than the other.

[Anatole Dedecker]

And (IIRC) none of them gives you the tensor product of Hilbert spaces (though I'm not sure it matters here, is there a significant theory of infinite dimensional Riemannian manifolds?)

[Oliver Nash]

Infinite-dimensional Riemannian manifolds are definitely important though at my end of differential geometry things were often quite hand-wavy when they came up. Many interesting finite-dimensional spaces can be constructed as infinite-dimensional quotients of infinite-dimensional Riemannian manifolds.

[Oliver Nash]

(I can supply references tomorrow if anyone wants them.)

[Michael Rothgang]

There is definitely Morse theory on infinite-dimensional Hilbert manifolds. (Though I'm not sure to what extent people caring about this will use convenient vector spaces, which are another rabbit hole in itself.)