Let $E$ be a real pre-Hilbert space. Show that in $E$ the mapping $x\to\||x||$ of $E$ into $\mathbf{R}$ is differentiable at every point $x\ne0$ and that its derivative at such a point is the linear mapping $$s\to\frac{(s\mid x)}{||x||}.$$ I am not sure what this $(s\mid x)$ means. However, I t...

Consider the map $f:\mathbb{R}^2\to\mathbb{R}^2$ defined as $$f(x,y)=\left(6x+(4+x^2+y^2/2)^{1/2},8y+\operatorname{log}(1+y^2)+\operatorname{arctan}(x)\right)$$ i) Prove that the preimage of a compact set through $f$ is compact ii) prove that $f$ is a homeomorphism from $\mathbb{R}^2 $ to $\math...

I'm trying to prove this assertion for arbitrary $f\colon U\subseteq E\to F$ where $U$ is an open in $E$ and $E,F$ are Banach spaces. In order to get some intuition, I tried proving it for the case $U,E,F=\mathbb{R}$ but can't think of a way that doesn't involve the fundamental theorem of calculu...