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".