# Space of Continuous on Closed Interval Real-Valued Functions with Supremum Norm forms Normed Vector Space

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## Theorem

Let $I := \closedint a b$ be a closed real interval.

The space of continuous real-valued functions on $I$ with supremum norm forms a normed vector space.

## Proof

We have that:

By definition, $\struct {\map C I, \norm {\, \cdot \,}_\infty}$ is a normed vector space.

$\blacksquare$

## Sources

- 2017: Amol Sasane:
*A Friendly Approach to Functional Analysis*... (previous) ... (next): $\S 1.4$: Normed and Banach spaces. Sequences in a normed space; Banach spaces