# Is there a nice “synthetic” way for doing differential geometry on infinite dimensional vector spaces?

If $V$ is an infinite dimensional vector space, for example the space of smooth functions on $\mathbb{R}$, we can introduce some differential geometry concepts by choosing a topology on $V$ and doing some fancy analysis.

I'd like to avoid this, since it seems unnecessary, while still having a meaningful, formal discussion. I'll try to explain my use of the word "synthetic" via some examples:

Let $V$ be a vector space over $\mathbb{R}$. I would want to say the following:

• A smooth function $V \rightarrow \mathbb{R}$ is a function which is smooth upon restriction to finite-dimensional subspaces.

• A smooth $k-$form on $V$ is a function $V \times (\Lambda^{k}V) \rightarrow \mathbb{R}$ which is smooth (in the sense of the previous bullet) and linear in the second coordinate.

• The tangent space $T_vV$ is isomorphic to $V$.

Already one can show quite easily that there is a differential $\delta$ from smooth $k-$forms to $k+1-$forms. If $A:V \rightarrow W$ is linear, one can pull-back smooth forms via $A$. There is a small issue with vector fields, since I'm not sure what a 'smooth vector field' should be. But at the very least, one can define an affine vector field to be an affine function $Q:V \rightarrow V$. For such vector fields, one recovers exterior multiplication $i_{Q}$ and one can prove the usual result that $i_{[X,Y]} = [\mathcal{L}_{X},i_{Y}]$. In short, the basic theorems of differential geometry follow naturally and easily.

My question is: Does something like this already exist in the literature? Is my description the "right thing" to do, or have people found better ways of discussing such concepts?

I should mention that my motivation is a simple, rigorous discussion of some of the basic concepts of field theory. In particular, a common set-up is a space of fields which is a vector space $V$ and an action functional which is a smooth function $S:V \rightarrow \mathbb{R}$, and the equations of motion $\delta S = 0$. It seems that one can get a lot of mileage and recover well-known results rigorously without introducing the words "Frechet Manifold".

• I would include the inverse function theorem in what you call "the basic theorems of differential geometry".In that settng even Fréchet manifolds are not sufficient to guarantee it holds... – Loïc Teyssier May 15 '15 at 20:16

## 2 Answers

Have a look at this Wikipedia page, or this book for the full story.

On the other hand, synthetic differential geometry includes infinite dimensional spaces. Also, diffeological spaces furnish what you want.

The paper Comparative smootheology give a nice overview of cartesian closed settings of smooth spaces.

Your first requirement suggests to me that you want to think of an infinite-dimensional vector space as an ind-object, namely the filtered colimit of its finite-dimensional subspaces. If so, one formal setting to try is ind-manifolds (formal filtered colimits of finite-dimensional smooth manifolds). People in algebraic geometry do analogous things when they view e.g. the affine Grassmannian as an ind-scheme. I don't know how this compares to other settings for synthetic differential geometry.

Ind-manifolds have tangent spaces which are ind-finite vector spaces (which are just vector spaces) but cotangent spaces which are pro-finite vector spaces, so differential forms in this setting might take some getting used to. And I don't know if the category of ind-manifolds is very well behaved.