Let $E$ be a vector space of all continuous functions $f\colon[0,1]\to\mathbb{R}$. Let $F$ be a subspace of $E$ that contains all functions $f$ in $E$ that have a continuous first derivative $f’$ in $]0,1[$ and so that $f’$ extends as a continuous function on $[0,1]$.
We have a norm on $E$, $\|f\|_\inf = \sup_{0\le x\le1} |f(x)|$. On $F$ we have two norms $\|f\|_\inf $ and $\|f\|_1 = \|f\|_\inf + \|f’\|_\inf$.
And finally we have a linear projection $T\colon F\to E$ given by $T(f)=f’$.
(i) Show that $T$ is continuous projection if we use the norm $\|f\|_1$ in $F$.
(ii) Show that $T$ is not continuous projection if we use the norm $\|f\|_\inf $ in $F$.
I know that a projection is continuous at a point $a$ if for every $\epsilon>0$ there exist $\delta >0$ such that $d_Y(f(x),f(a))<\epsilon$ for all $x\in X$ that meet $ d_X(x,a)<\delta$.
And than a projection is continuous if it’s continuous at every point $a$.
But I’m not really sure how to use the norms in this and where.
I was doing something like this to show it’s continuous at some $a$.
$f_n \rightarrow v$, then $\|f_n \rightarrow v\|\rightarrow a$
so $f_n-v\rightarrow a$
Than we get $T(f_n-v)\rightarrow a$
$T(f_n)-T(v)\rightarrow a$
$T(f_n)\rightarrow a-T(v)$
But I don’t know if that’s enough and I don’t think I’m even using the norms so I’m pretty lost. If anyone has some ideas it would really help.