# 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.

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