Open and closed balls in $C[a,b]$

Question

Let $X$ be a non empty set and let $C[a,b]$ denote the set of all real or complex valued continuous functions on $X$ with a metric induced by the supremum norm.

How to find open and closed balls in $C[a,b]$? Can we see them geometrically? For example what is an open ball $B(x_0;1)$ i.e. ball centered at $x_0$ with radius $1$ in $C[a,b]$. I can visualize them in $\mathbb R^n$ but when it comes to functional spaces I have no clue how to identify them?

Thanks for helping me.

Answer 1

Yes, you should think of it just like you think of any other metric space. Every norm $\|\cdot\|$ induces a metric $d(x,y) := \|x-y\|$.

In your example, $$ B(x_0, 1) = \{ f: X \to \mathbb R \Big \vert \|f - x_0\|_\infty < 1 \}$$

In the $\sup$-norm, these $f$ are all functions that are never further away from $x_0$ at any given $x$ in $\mathbb R$. This is what it looks like: