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The resulting metric topology corresponding to the norm given by: $\|f\| = \sup\limits_{x\in X} | f(x) |$ on $C^*(X)$ is called the uniform norm topology on $C^*(X)$. Show that, in uniform norm topology is uniform convergence of the functions.

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    @SugataAdhya please learn to use MathJax for the mathematics. When one is accustomed to reading things like "\lvert F_n(f) - F(f)\rvert < \varepsilon", seeing it rendered as "| Fn(f) – F(f)| < Є" is slightly confusing. It's also a simple fact that people take questions with properly formatted mathematics (and spelling and grammar—not an issue for you) a lot more seriously.2012-11-17

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As I understand it, you are trying to show that if $C^*(X)$ is given the uniform norm topology, then if a sequence of functions on $C^*(X)$ converges, then it converges uniformly. So you started out taking a sequence $F_1, F_2,...$ of elements of $C^*(X)$, but these are functions of $X$, so you should not have written $F(f)$ where $f$ is in $C^*(X)$. But if you fix that mistake, then your answer is right. This question is really just a matter of getting the definitions right.

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    Yes, that's right.2012-07-26