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The following theorems are equivalent? Is the Theorem 2 false?

Theorem 1 (Arzela-Ascoli): Let $X$ be a compact metric space and let $C(X)$ denote the space of continuous functions $f: X \to \mathbb C$ endowed with the sup norm $\|\cdot \|_\infty$. Then $S \subseteq C(X)$ is relatively compact if and only if it is pointwise bounded and equicontinuous.

Theorem 2: Let $X$ be a compact metric space and let $C(X)$ denote the space of continuous functions $f: X \to \mathbb C$ endowed with the sup norm $\|\cdot \|_\infty$. Then $S \subseteq C(X)$ is relatively compact if and only if it is pointwise bounded at some point and equicontinuous.

Any help would be appreciated.

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The two variants are not equivalent, and "Theorem 2" is false.

Consider $X = \{0,1\}\subset \mathbb{R}$, and $S = \{ f_z : z \in \mathbb{C}\}$, where $f_z(p) = p\cdot z$. Then $S$ is equicontinuous and bounded at $0$, but not relatively compact.

"Theorem 2" becomes true, and the two theorems equivalent if $X$ is additionally required to be connected, for under that condition, the boundedness at one point implies the uniform boundedness on the entire space.

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  • $\begingroup$ I see it now by the connectedness. Thanks for your solution! $\endgroup$ – felipeuni Nov 2 '14 at 0:41

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