Let $C^1[0,1]$ be space of all real valued continuous function which are continuously differentiable on $(0,1)$ and whose derivative can be continuously extended to $[0,1]$. For $f$ in that set define the norm of $f = \max{\{\|f(t)\|, \|f'(t)\|\}}$ where $t$ is in $[0,1]$. can you show this space is complete under this norm? and let a linear operator $T:C^1[0,1] \longrightarrow C[0,1]$ defined by $T(f)=f'$, show that $T$ is continuous and norm of $T=1$.
[Math] Space of all continuously differentiable functions
banach-spacesfunctional-analysis
Best Answer
You probably mean $\|f\| =\max{\{\|f\|_{\infty},\|f'\|_{\infty}\}}$ where $\|f\|_{\infty} = \sup_{t \in [0,1]} |f(t)|$.
Hint: Recall the basic fact that if $f_{n}$ is a sequence of $C^{1}$ functions such that $f_{n}$ converges uniformly (or only pointwise) to a continuous function $f$ and $f_{n}'$ converges uniformly to a continuous function $g$ then $f \in C^{1}$ and $f' = g$. It's probably a good idea to try to prove this on your own without consulting your books.
That $\|T\| \leq 1$ is just the definition of the operator norm. Consider also $f(x) = x$.