I have been thinking lately, as I am learning more about hilbert spaces and some functional analysis, about the following definition;

Suppose $(V,||.||_v)$ and $(W,||.||_w)$ are normed spaces and let $f:V \rightarrow W$ be a function. Then consider the function $g:V \rightarrow \mathbb{R}$, defined by

\begin{align*} g(v):= \lim_{||h||_V\rightarrow 0}\frac{||f(v+h)-f(v)||_W}{||h||_V}. \end{align*}

We see that this is very similar to the usual definition of the derivative of the usual single variable functions.

I understand how the derivative of functions of multiple variables work, but I am wondering if this function $g$ makes any real sense if we apply it to say function spaces, matrix spaces and so on.

If this is a well documented idea, please tell me what I can search in google or on Stack Exchange to learn more. If this function is nonsense, or not even a function at all, please tell me explain a bit more.


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