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

From ProofWiki
Jump to navigation Jump to search

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:

Space of Continuous on Closed Interval Real-Valued Functions with Pointwise Addition and Pointwise Scalar Multiplication forms Vector Space
Supremum Norm is Norm/Continuous on Closed Interval Real-Valued Function

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

$\blacksquare$


Sources