Clarification on how a particular space is defined

Question

I am tasked with proving that the the metric space $(C[0,1], d)$ is complete where $d$ is the infinity-metric.

As of right now, I interpret the set $C[0,1]$ as the set of continuous functions from $0$ to $1$, but I find the idea of calculating a distance between elements in this space to be very abstract. Could someone tell me what the infinity-metric means when applied to the set $C[0,1]$, or at least provide me with some sort of clue?

Answer 1

It means that $$ d(f,g)=\sup_{x\in[0,1]}|f(x)-g(x)| $$ In particular $f_n\to f$ in this metric means precisely that $f_n\to f$ uniformly.

Answer 2

Foobaz John tells what the metric in $C[0,1]$ is. Also that metric comes from the norm $$\vert\vert f \vert \vert=\sup_{x \in C[0,1]} \vert f(x) \vert$$

Also you understand what they are, see this picture: