I started new in funcional analysis and have a question about Banach Spaces. I'm supposed to show that
$X = C^0([0,1]) = {f:[0,1] \rightarrow \mathbb{C}, f \quad \text{continous} } $
is a Banach Space with the Supremum Norm
$||f||_{C^0} = ||f||_\infty = \sup_{x \in [0,1]} |f(x)|$.
I've shown Banach Space with limited functions, but I'm not sure how to work here, because $f$ can diverge.