Finding the Norm of an Element

Question

This may sound very trivial, but I do not know what I am missing.

Take $X$ be the space of complex valued continuous functions on $[0,1]$ with the usual sup norm. Take $Y=\{f \in C[0,1]:f(0)=0\}$.

Show that:

1. $Y$ is closed in $X$
2. $X/Y$ is isomorphic to the set of complex numbers $\mathbb{C}$.
3. Consider the quotient space $X/Y$ with the usual quotient norm. Let us denote the elements of $X/Y$ by $[x]$. Show that $||[f]||=|f(0)|$.

I have done 1 and 2, but unable to do 3.

Thanks for any help.

Answer 1

$[f]$ contains the constant function $f(0)$ and $$\|[f]\|_{X/Y} = \inf_{g \in [f]} \max_{x \in [0,1]} |g(x)| \ge \inf_{g \in [f]} |g(0)| = |f(0)|.$$