Supremum Metric on Continuous Real Functions/Examples

Examples of Supremum Metric on Continuous Real Functions

Closure of $\map {B_1} 0$ on $\closedint 0 1$

Let $\closedint 0 1 \subseteq \R$ be the closed unit interval.

Let $\mathscr C \closedint 0 1$ be the supremum space of continuous functions $f: \closedint 0 1 \to \R$.

Then:

$\map \cl {\map {B_1} \bszero} = \set {f \in \mathscr C \closedint 0 1: \map {d_\infty} {f, \bszero} \le 1}$

where:

$\map {B_1} \bszero$ denotes the open $1$-ball of $\bszero$

$\d_\infty$ denotes the Chebyshev distance

$\bszero$ denotes the constant function $f_0$.