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While reading through Kantorovitz's book on functional analysis, I had a query that need clarification. If $X$ is compact, $C_{B}(X)$ - bounded continuous function, with the sup-norm coincides with $C(X)$ - continuous real valued function, with the sup-norm, since if $f:X \rightarrow \mathbb{R}$ is continuous and $X$ is compact, then $\vert f \vert$ is bounded.

May I know how the above relates to the corollary that states: Let $X$ be a compact topological space. If $f \in C(X)$, then $\vert f \vert$ has a minimum and a maximum value on $X$. I believe the relation here is that the function is bounded and hence relate to the corollary but hope someone can clarify just to be sure. Thank You.

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up vote 1 down vote accepted

That is exactly what you said, but changing it a little:

Every continuous real function over a compact space is bounded.

We know that the image of a compact set by a continuous function is compact, and that implies boundedness of the image. A function is bounded exactly when its image is bounded, so it's proved!

Then $C(X)=C_B(X)$. The minimum and maximum is a plus, that implies boundedness, so that your reasoning was correct.

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I see. Thanks for the clarification, Juan. ;p – Sandra May 29 '12 at 2:48

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