Closure w.r.t $|| . ||_\infty$ Let $A=C[0,1]$ and $B=\{f\in A|f(0)=0\}$. What is the closure of B w.r.t the $|| . ||_\infty$ norm.
I really have a very poor intuitive grasp of this problem and I'm not sure where to begin?
I know $|| f ||_\infty=\sup_{x\in[0,1]} f(x)$ and that the closure is the limit points of B 
Thansk for any help

The map $L\colon A\to \Bbb R$, $f\mapsto f(0)$ is linear, and continuous as $|L(f)|=|f(0)|\leq \lVert f\rVert_{\infty}$. Since $B=L^{-1}(\{0\})$, $B$ is closed hence equal to its closure.
