Terminology of a space in real-analysis

Question

Since this is a very simple question, i didn't want ask this here not to bother you, so i saw wikipedia and googled this but still don't get what this space is called..

I want to know (i)name, (ii)abbreviation and (iii)Latex of this space. (i.e. (i)Real Number (ii)$\mathbb{R}$ (iii)$\text{ \mathbb{R}}$)

Here it is;

$A(X)$ is a set of all complex-valued, continuous, bounded functions with domain $X$ where $X$ is a metric space.

What is this $A(X)$ called? Thank you in advance..

Usually $C(X)$ denotes the continuous $\Bbb C$ valued functions, I'm not aware of any specific name, and somehow you should also express boundedness, it is usually done by a subscript 'b'. — Berci, Dec 14 '12 at 10:21

score 3 · Accepted Answer · answered Dec 14 '12 at 10:25

"Continuous" only refers to the topology so there's no reason to restrict yourself to the case of metric spaces. My understanding is that the standard notation is $C_b(X)$ ($C$ for continuous, $b$ for bounded, and the complex is implied). With the sup norm this is a C*-algebra, isomorphic via the Gelfand representation to the C*-algebra $C(\beta X)$ of all continuous complex-valued functions on the Stone-Čech compactification of $X$.

Terminology of a space in real-analysis

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