Proof of the Chain Rule in not normed TVS

Question

In Differentiable and Riemannian manifolds by Lang, the chain rule theorem for TVS is stated, but the proof is left as an exercise. I gave it a try, so I expose here my idea. In order to check if I am on the right track, or if there is something I am missing.

In the textbook, the definition of differentiable map is the following:

Def. Let $E,F$ be two topological vector spaces and $U\subseteq E$ and open set, a function $f:U\rightarrow F$ is said to be differentiable at $x_0\in U$ if there exists a linear map $\lambda:E\rightarrow F$ such that if we let $$f(x_0+y)=f(x_0)+\lambda y+\psi(y) $$ for small $y$, then $\psi$ is tangent to $0$

First of all I want to express my discomfort with the phrase for small $y$, which seems pretty arbitrary, and really confuses me a bit. Secondly, by tangent to $0$ the author means that for any neighborhood $W$ of $0\in F$, there exists a neighborhood of $0\in E$ such that $$\psi(tV)\subseteq o(t)W $$ where $o$ is real valued function of a real variable such that $\text{lim}_{t\rightarrow 0}o(t)/t=0$

The main theorem is the classical one

Theorem: If $f:U\rightarrow V$ is differentiable at $x_0\in U$ and $g:V\rightarrow W$ is differentiable at $f(x_0)$, then $g\circ f$ is differentiable at $x_0$ and $$(g\circ f)'(x_0)=g'(f(x_0))\circ f'(x_0) $$

I figured this must be analogous to the usual proof of the chain rule. So, here is my attempt of a proof:

Proof: We have that $$g'(f(x_0))y=g(f(x_0)+y)-g(f(x_0))-\psi_g(y)$$ by the definition of differentiablity (for small $y$, whatever that means). In particular, take $y=f'(x_0)z+\psi_f(z)$ for some small $z$, where $\psi_f$ is the tangent to $0$ map form the definition of differentiability of $f$. Our expression is now $$[g'(f(x_0))\circ f'(x_0)](z)+[g'(f(x_0))\circ \psi_f](z)= g(f(x_0+z))-g(f(x_0))-[\psi_g\circ f'(x_0)](z)-[\psi_g\circ \psi_f](z)$$ so it suffices to show that for the small $z$ the maps $g'(f(x_0))\circ \psi_f,\;\psi_g\circ f'(x_0)$ and $ \psi_g\circ \psi_f$ are tangent to $0$, the last map is trivially tangent to $0$. But I am clueless about proving that other two are in fact tangent to $0$

Answer 1

In fact, there are difficulties in defining a correct differentiability notion in general TVS. I invite you to consider Gateaux derivative and Differentiation in Fréchet spaces and the paragraph ``Tame Fréchet spaces'' for the last one.

Bourbaki himself in his volume Differentiable and Analytic Manifolds constructs his theory on Banach spaces.

As far as Lang's book is concerned, you can see that after Proposition 3.1, he returns to Banach spaces All topological vector spaces are assumed to be Banach spaces, so there is no harm to comment your proof in this context.

Now, returning to your proof, I would not begin by a difference (because along the way, you will encounter cones for which the increase has to be positive).

Using your notations, we can write $$ f(x_0+x)=f(x_0)+f'(x_0).x+\psi_f(x)\mbox{ and } g(y_0+y)=g(y_0)+g'(y_0)y+\psi_g(y) $$ the $\psi$ functions being in the little-o class (this time - hopefully - correctly defined) i.e. $$ \psi_f(x)=||x||.\epsilon_f(x)\mbox{ and } \psi_g(y)=||y||.\epsilon_g(y) $$ with $\lim_{h\to 0}\epsilon_i(h)=0$.

Then the proof follows classically (but I can give full details if requested).

Hope it helps.

Note Your two questions (this one and that one) are based on the (yet very interesting) book [1], please find below an analysis of the contexts and reasons.

Framework

Chapter 1.2 (Topological vector spaces)
1. [First page] Spaces are Hausdorff and locally convex.
2. [Next page] For this book, we assume from now on that all our topological vector spaces are Banach spaces.

Linear approximation

1. Chapter 1.3 _Derivatives and Composition of Maps_, S. Lang starts with an extension to general TVS of the notion of [Linear approximation](https://en.wikipedia.org/wiki/Linear_approximation) as follows
2. let $E,F$ be two TVS, he defines a function $\varphi$ from a neighbourhood of zero in $E$ to a neighbourhood of zero in $F$ to be tangent to $0$ if, for all neighbourhood $W$ of $0$ in $F$, there is a neighbourhood $V$ of $0$ in $E$ (I suppose $V\subset Dom(\varphi)$) such that, for small $t$ $$ \varphi(tV)\subset \psi(t)W $$ where $\psi$ is some function in the $o(t)$ class (just below).
3. he (re)defines, like Laudau, the $o(t)$ class, i.e. real functions of a real variable such that $\lim_{t\to 0}\psi(t)/t=0$. Note that such a function $\psi\in o(t)$ is assumed to vanish at zero and can equivalently be written $\psi(t)=t\varepsilon(t)$ where $\lim_{t\to 0}\varepsilon(t)=0$

This definition boils down to the classical one in case of Banach spaces.

[1] Serge Lang, Differential and Riemannian Manifolds, Graduate texts in mathematics (160), Springer (1995)