Non-standard analysis offers very convenient tools to prove facts about continuity or differentiability. I am looking for such tool in infinite-dimensional calculus.
To be more precise, let $X$ and $Y$ be Banach spaces and let ${}^*\!X, {}^*Y$ be their non-standard extensions. Suppose we have function $f\colon X\to Y$ and $x_0\in X$. What is the non-standard statement about ${}^*\!f$ corresponding to saying that $f$ is Frechet-differentiable at $x_0$?
On
There's nothing tricky about this; you just translate the limit into a statement about infinitely close points like you always do. A bounded linear map $A:X\to Y$ is the Frechet derivative of $f$ at $x_0$ if whenever $h\in {}^*X$ is infinitesimal, $f(x_0+h)=f(x_0)+Ah+\epsilon$, where $\|\epsilon\|/\|h\|$ is infinitesimal.
There doesn't seem to be any novelty as compared to the usual definition. Namely, let's say that $\alpha$ is infinitesimal if it is in the halo of the origin. Then linear map $A:X\to Y$ is the Frechet derivative of $f$ iff $|f(x_0+\alpha)-f(x_0)-A(\alpha)|_Y$ is infinitesimal compared to $|\alpha|_X$ for each infinitesimal $\alpha$. Here the halo of $0$ is the intersection of all the natural extensions of open neighborhoods of $0$.