- #1

camillio

- 74

- 2

In Baby Rudin, Theorem 7.25 states:

If [itex]K[/itex] is compact, [itex]f_n \in C(K)[/itex] for [itex]n=1,2,3,...[/itex] and if [itex]{f_n}[/itex] is pointwise bounded and equicontinuous on [itex]K[/itex], then

(a) [itex]{f_n}[/itex] is uniformly bounded on [itex]K[/itex]

The theorem continues with point (b), which I understand.

My question is, whether point (a) needs the equicontinuity, since:

* [itex]C(K)[/itex] is a set of continuous bounded functions on a compact space, i.e. uniformly continuous, hence there is no function [itex]g[/itex] s.t. [itex]g(x) \to \infty[/itex].

* if the previous holds, then by the hypothesis of pointwise boundedness of [itex]\{f_n\}[/itex], there must exist a number [itex]\sup_{x \in K, n=1,2,3,...} |f_n(x)| = M<\infty[/itex], [itex]x\in K[/itex] arbitrary and hence the uniform boundedness should hold.

Am I wrong?

If [itex]K[/itex] is compact, [itex]f_n \in C(K)[/itex] for [itex]n=1,2,3,...[/itex] and if [itex]{f_n}[/itex] is pointwise bounded and equicontinuous on [itex]K[/itex], then

(a) [itex]{f_n}[/itex] is uniformly bounded on [itex]K[/itex]

The theorem continues with point (b), which I understand.

My question is, whether point (a) needs the equicontinuity, since:

* [itex]C(K)[/itex] is a set of continuous bounded functions on a compact space, i.e. uniformly continuous, hence there is no function [itex]g[/itex] s.t. [itex]g(x) \to \infty[/itex].

* if the previous holds, then by the hypothesis of pointwise boundedness of [itex]\{f_n\}[/itex], there must exist a number [itex]\sup_{x \in K, n=1,2,3,...} |f_n(x)| = M<\infty[/itex], [itex]x\in K[/itex] arbitrary and hence the uniform boundedness should hold.

Am I wrong?

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