Differential of a function in non-normed topological vector spaces

Question

I would like to know if it is possible to define the differential of a function in a topological vector space that does not have a norm. To make things clear, let $E$ and $F$ two topological vector spaces over $\mathbb{R}$, such that there is a norm $\left\|\cdot\right\|$ defined on $E$, let $U$ an open set of $E$, $f:U\to F$ a function from $U$ to $F$ and $u\in U$. As the function defined by $\left\{\begin{array}{c}E\to E\\h\mapsto u+h\end{array}\right.$ is continuous, there exits a neighbourhood $V$ of $0$ in $E$ such that $\forall h\in V,\,\left(u+h\right)\in U$, such that for $h\in V$, the value $f\left(u+h\right)$ is defined. Then, the differential of function $f$ at point $u$ is generally defined as the only linear map $\text d_uf:V\to F$ such that:

$$f\left(u+h\right)=f\left(u\right)+\text d_uf\left(h\right)+\underset{h\to0}{o}\left(h\right)\text{ ;}$$

where $\underset{h\to0}{o}\left(h\right)$ is Landau's small o notation, which means:

$$\lim\limits_{h\to0}\frac{f\left(u+h\right)-f\left(u\right)-\text d_uf\left(h\right)}{\left\|h\right\|}=0\text{.}$$

It is clear that the latter definition involves the norm $\left\|.\right\|$ defined on $E$. My question is, can we define the differential of a function (or an equivalent of Landau's notation) such that it does not involve the norm $\left\|\cdot\right\|$, so that it is defined on any topological vector space - i.e. not necessarily one which has a topology induced by a norm.

Thank you for any answer.

Answer 1

See page 6 of S.Lang, "Differential and Riemannian manifolds". He defines differentiability (in the sense you are asking, which is stronger than Frechet) for general topological vector spaces.

Definition. Let $E, F$ be topological vector spaces and $\phi: U\subset E\to F$ is a map of a neighborhood $U$ of $0\in E$. Then $\phi$ is said to be tangent at $0$ if for every neighborhood $W$ of $0\in F$ there exists a neighborhood $V$ of $0$ in $E$ such that $\phi(tV)\subset o(t) W$ for some function $o(t)$ (little $o$ in the usual sense of functions of one real variable).

Definition. A map $f: E\to F$ is said to have derivative $L$ at $a\in E$ if $L: E\to F$ is a continuous linear map such that (for $x$ in some neighborhood of $0\in E$), $$ f(a+x)= f(a) + L(x) + \phi(x) $$ for some $\phi$ tangent at $0$.

Answer 2

Frechet derivative is equivalent to Caratheodory derivative in Banach spaces. we define a map f : X --> y is differentiable at a point x, if there exists a continuous linear map g from X to Y such that f( v) - f(x) = g (v-x) for all v in a neighborhood V of x . The value of the derivative is g ( x ). The map g depends on the point x Ah laws of derivatives including the chain rule hold. One can prove implicit function theorem for a map

F X * Y to Z where X is a TVS and Y , Z are Banach spaces. provided the partial derivative of F with respect to y exists and is invertible as a continuous linear map : Z --> Y .

The proof is essentially same that of Dieuodenne "Foundations of Modern analysis". For existence and continuity X can be just topological space. The differentiability part holds with minor modifications when X is TVS.. it also holds with Lang's definition of derivative but lesser transparent.

The details when X is a metric space are already in a textbook of Topology.