How prove this set isn't dense? I want to prove this set $M=\{U \in X ,\  \|U\|\le 1\}$  isn't dense in $X =C[a,b]$.
Can you help me?

The set is not dense in $C([a,b])$. This means that the closure of $M$ does not equal $C([a,b])$. To see this, let $f\in C([a,b])$ such that $\|f\|=2$. Take the open ball of radius $1/2$ centered at $f$, $B(f,1/2)$. So if $g\in B(f,1/2)$ we have that $3/2< \|g\|< 5/2$. Hence, $g\notin M$ and so $B(f,1/2)\cap M=\emptyset$. Since there exists an open neighborhood of $f$ whose intersection with $M$ is empty, we have that $f$ is not in the closure of $M$. Thus, the closure of $M$ is not equal to $C([a,b])$ and so $M$ is not dense in $C([a,b])$.
