Let $T \in L(\mathbb{R}^n, \mathbb{R}^m)$. Prove that $T$ is differentiable at every $x \in \mathbb{R}^n$.
Thinking to start with the fact that $T = Ax$, for some $m × n$ matrix $A$, but not sure where to go with that.
Lost in the notation.
I think I need to demonstrate $$\lim_{H \to 0} \frac{f(x_0 + H)-f(x_0)-T(H)}{||H||} = 0.$$
Please help steer me. 1st round of analysis in $\mathbb{R}^n$.
Here, we have $f(x) = T(x)$. In the notation you're using, $f':\Bbb R^n \to L(\Bbb R^n, \Bbb R^m)$, which is to say that we should have $f'(x) \in L(\Bbb R^n,\Bbb R^m)$ for every $x \in \Bbb R^n$. In fact, it turns out that we should have $f'(x) = T$ for every $x \in \Bbb R^n$; consider the fact that your linear function has a constant derivative (i.e $f'(x)$ should be the same linear transformation for all $x$).
So in particular, what you need to show is that $$ \lim_{h \to 0} \frac{f(x + h) - f(h) - Th}{\|h\|} = 0 $$ This is very easy to show, since the numerator of that fraction is always $0$.