Definition:Space of Continuous Functions on Compact Hausdorff Space
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Definition
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 $\map \CC X$.
Also see
Sources
- 1991: Walter Rudin: Functional Analysis (2nd ed.) ... (previous) ... (next): $3.13$