Properties needed to define Derivatives on Topological space I just started learning topology and was curious about defining derivatives on general topological spaces.
Since we can define continuous functions on Topological spaces, my question is what additional properties one would need to define derivatives on Topological spaces.
I guessed one might only need converging sequences to define derivative, so space must have,

*

*metrizability: to define some sort of distance between points so one can define converging sequences on space.

*Hausdorff property: so convergence would be unique.

but on the other hand, these properties are conserved under homeomorphism, where differentiability does not.
so, there should be some more properties (maybe other than topological properties) one would need to define derivatives, and I'm not sure what kind of property it would be that conserves differentiability.
I hope my question makes sense.
Thanks.
 A: The derivative of a function $f$ at a "point" $a$, in the classical sense, is the limit of the "size" of $f(a+h)-f(x)$ multiplied by the real $1/h$.
This needs addition/subtraction and multiplication by a real number to be defined, and a norm (for managing the "size"). Therefore, an adequate framework is that of a normed topological vector space, and even a complete normed topological space (see Remark 3 below) which is called a Banach space.
Remarks :

*

*Were you interested by the extension of integration (which has strong connection with differentiation in our usual spaces), you need to study measure theory, whose axioms bear a certain similarity with those of a topological space (both forged in the 1930's). A very long (and deep!) paper by Field medallist Terence Tao about these issues (maybe difficult for you, at an early stage of your studies) is this one : https://terrytao.files.wordpress.com/2010/02/epsilon.pdf


*Instead of a norm we could have used a distance : such a space is called a Fréchet space.


*Recall : a space is said "complete" if the concept of Cauchy sequence is the same as the concept of converging sequence ; asking for completeness is understandable because it is under this hypothesis that the so important geometric series $\sum a^k$ with $\|a\|<1$ are convergent.
A: The domain and range of your functions should have a linear structure to begin with. Then, to talk about limits, your space should have a topology. Topological linear spaces are suite for this.

In normed spaced (Banach spaces for example) there is a notion of total derivative that generalizes the concept of derivative known in Calculus.
Definition (Fréchet) Suppose $X,Y$ are normed spaces, $U\subset X$ open. A function $F:U\longrightarrow Y$ is called differentiable at $x\in U$ if there is $F'(x)\in\mathcal{L}(X,Y)$ such that
$$
F(x+h)=F(x)+F'(x)h + r(h)
$$
where  $r(h)=o(h)$; i.e., $\lim_{h\rightarrow0}\frac{|r(h)|}{\|h\|}=0$.

This notion does not generalize to locally convex spaces.

There is another option of differentiability that focusses on directional derivatives which can be generalize to general linear topological spaces, in articular locally convex spaces.
Definition: Suppose $X$ and $Y$ are locally convex linear spaces, $U\subset X$ open. Let function  $F:U\rightarrow Y$.  The directional derivative of $F$ at $\mathbf{x}\in U$ in the direction $\mathbf{v}$ is defined as
$$
D_vF(\mathbf{x}):=\lim_{t\rightarrow0}\frac{F(\mathbf{x}+t\mathbf{v})-F(\mathbf{x})}{t}
$$
when the limit exists (the limit is with respect to the topology in $Y$, that is for any neighborhood $U$ of $\mathbf{0}\in Y$, there is $\delta>0$ such that if $0<|t|<\delta$, $\frac{F(\mathbf{x+v})-F(\mathbf{x})}{t}\in D_vF(\mathbf{x})+ U$).
Definition: $F$ is said to be Gâteaux--differentiable at $\mathbf{x}\in U$ there is a map $L_x:X\rightarrow Y$ such that $D_vF(\mathbf{x})=L_x\mathbf{v}$ for all $\mathbf{v}\in X$. $L_x$ is called Gâteaux--derivative of $F$ at $\mathbf{x}$.

If $X$ and $Y$ are Banach spaces, and
$F$ is differentiable at $\mathbf{x}\in U$,  then  $F$ is Gâteaux--differentiable at $\mathbf{x}$ and
$D_vF(\mathbf{x})=F'(\mathbf{x})\mathbf{v}$ for all $\mathbf{v}\in X$. The converse is not necessarily true, unless some continuity conditions on $x\rightarrow L_x$ are satisfied.
Theorem:
Suppose $X$ and $Y$ are Banach spaces, $U\subset X$ open, and let $F:U\subset X\rightarrow Y$ be Gâteaux--differentiable on a neighborhood $V\subset U$ of a point $\mathbf{x}\in U$. If the Gâteaux derivative $y\mapsto L_y\in \mathcal{L}(X,Y)$ \is continuous at $\mathbf{x}$, then $F$ is (Fréchet) differentiable at $\mathbf{x}$ and $F'(\mathbf{x})=L(\mathbf{x})$.

Many books in nonlinear functional analysis (Klaus Deimling's for example) have more details and application of Gâteaux differentiation.
