0
$\begingroup$

this is theorem 7.24 of rudin's principle of mathematical analysis:

If $K$ is a compact metric space, if $f_n \in C(K)$ for $n=1,2,3,...$ ($C(K)$ being a set of complex-valued, continuous and bounded functions), and if $\{f_n\}$ converges uniformly on $K$, then $\{f_n\}$ is equicontinuous on $K$.

I was wandering about necessity of $K$ being compact.

can anyone help me with an example of a sequence of functions $\{f_n\}$ on $C(K)$( which $K$ is not a compact metric space) such that $\{f_n\}$ converges uniformly on $K$,but then $\{f_n\}$ is NOT equicontinuous on $K$.

$\endgroup$
1
$\begingroup$

Well, if $K$ isn't compact then its perfectly possible that $f_1$ is not uniformly continuous. If you add the condition that every function be uniformly continuous the theorem still holds.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.