Relation between min max of a bounded with compact and continuity 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.

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.
