2
$\begingroup$

Let $X$ be a topological vector space with the first countable topology(that is, every point has a countable neighborhood basis).If every Cauchy sequence convergence, we want to show that every Cauchy net converges.

For every given Cauchy net $\langle x_i\rangle$, I think I can inductively find a subnet $\langle x_{k_n}\rangle$of $\langle x_i\rangle$ which is actually a Cauchy sequence. By assumption, we know $\langle x_{k_n}\rangle$ converges to a point $x$ in $X$, I need to show $\langle x_i\rangle \rightarrow x$.

If $X$ is a metric space, then the argument may be easier since we can apply "$2\epsilon$ argument" to $x_i-x_{k_n}+x_{k_n}-x$. But how to do this in a general 1st countable topological vector space? Plus, if $\{U_n\}$ is a countable neighborhood basis at $0$, is $\{U_n+U_n\}$ again a neighborhood basis at $0$?

$\endgroup$
2
$\begingroup$

if $\{U_n\}$ is a countable neighborhood basis at $0$, is $\{U_n+U_n\}$ again a neighborhood basis at $0$?

Yes. Let $U$ be an arbitrary neighborhood of the zero. There exists a neighborhood $V$ of the zero such that $V+V\subset U$, and there exists a base neighborhood $U_n$ of the zero such that $U_n\subset V$. This should allow you to apply "2$\epsilon$ argument"

| cite | improve this answer | |
$\endgroup$

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.