# 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... May 15, 2015 at 20:16