I am reading some literature where they refer to
"the Banach space of all $C^1$ maps from $[0,1]$ to $\mathbb{R}^n$."
But they don't say what norm they are using to make this into a Banach space. Is there a natural/standard choice of norm on this space which works?
My guess is that $$\|f\|=\sup_{x\in[0,1]}|f(x)|+\sup_{x\in[0,1]}|f'(x)|.$$ But I am not sure if this gives a Banach space. Is it? (Additionally, I would be interested to see a textbook reference of this material.)
Say you have a Cauchy sequence $\{f_n\}$ in the $C^1$ norm you propose. Then in particular $\{f_n\}$ is Cauchy in $C^0$ and so converges uniformly to a continuous function $f$. Likewise $\{f_n'\}$ is Cauchy in $C^0$ and converges uniformly to a continuous function $g$. Since $$f_n(x) = f_n(0) + \int_0^x f_n'(t) \, dt,\quad x \in [0,1]$$ holds for all $n$, you can let $n \to \infty$ and get $$f(x) = f(0) + \int_0^x g(t) \, dt,\quad x \in [0,1].$$ Thus $f \in C^1$ and $f' = g$ so that $f_n \to f$ in your $C^1$ norm.
