Supremum Metric on Continuous Real Functions/Examples
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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 $f_0$.