Category:Space of Continuous Functions on Compact Hausdorff Space

This category contains results about Space of Continuous Functions on Compact Hausdorff Space.
Definitions specific to this category can be found in Definitions/Space of Continuous Functions on Compact Hausdorff Space.

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 $\map \CC X$.