Definition:Space of Continuous Functions on Compact Hausdorff Space

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Let $X$ be a compact Hausdorff space.

Let $\struct {Y, \norm {\, \cdot \,}_Y}$ be a Banach space.

Let $\CC = \CC \struct {X; Y}$ be the set of all continuous mappings $X \to Y$.

Equip $\CC$ with the vector space operations inherited from the vector space $Y^X$.

Define $\norm {\, \cdot \,}_\infty : \CC \to \R$ by:

$\ds \norm f_\infty = \sup_{x \in K} \norm {\map f x}_Y$

We call $\struct {\CC, \norm {\, \cdot \,}_\infty}$ is the space of continuous functions on $X$ valued in $Y$.

If $Y = \C$, we may write $\map \CC {X; Y}$ as simply $\map \CC X$.

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