5
$\begingroup$

I study about weak and weak* topology in functional analysis.

By Eberlein-Smulian, every weakly compact set is weakly sequentially compact. How about weak* topology? I learned that $(B_{X^*},\omega^*)$($\omega^*$ means weak* topology.) is metrizable when $X$ is separable, so it is clearly true for $(B_{X^*},\omega^*)$, but I don't know the result for $(X^*,\omega^*)$.

On the other hand, does this hold about general topology? i.e., if $(X,\tau_1)$ is a topological space that $\{K\subset X:K$ is compact$\}$=$\{K\subset X:K$ is sequentially compact$\}$ and $(X,\tau_2)$ is a coarser topology than $\tau_1$, does the same hold for $(X,\tau_2)$? I think it is false but cannot find examples.

$\endgroup$
  • $\begingroup$ The weak$^*$ topology on $X^*$ is metrizable if $X$ is separable, but not in general. $\endgroup$ – Aweygan Jan 11 '17 at 9:31
  • $\begingroup$ I confused something about weak* topology. Thanks a lot! $\endgroup$ – CSH Jan 11 '17 at 9:33
  • $\begingroup$ You're welcome. The last statement of my comment was incorrect. I confused the content of the Banach-Alouglu theorem to be about the weak topology. Nevertheless, it is still relevant. $\endgroup$ – Aweygan Jan 11 '17 at 9:37
  • $\begingroup$ Ok! Have a good day $\endgroup$ – CSH Jan 11 '17 at 9:38
  • $\begingroup$ Here is a related post. One can show that for $X=\ell_1(\Bbb R)$, $B_{X^*}$ is not weak* sequentially compact. $\endgroup$ – David Mitra Jan 11 '17 at 10:14
1
$\begingroup$

The weak* topology of $X^*$ is never sequential, unless $X$ is finite-dimensional. To see this , you may modify this proof.

$\endgroup$
  • $\begingroup$ That post asks whether or not every weakly sequentially closed set is weakly closed. This post asks whether or not weak$^*$ sequential compactness is equivalent to weak$^*$ compactness, which is the case when $X$ reflexive. $\endgroup$ – Aweygan Jan 12 '17 at 20:46
  • $\begingroup$ How can I relate sequentially closed and sequentially compact? Are they equivalent? $\endgroup$ – CSH Jan 13 '17 at 0:02
  • $\begingroup$ @Aweygan, there are two questions: the first one is about metrasibility/sequentiality of $X^*$ in the weak*-topology. $\endgroup$ – Tomek Kania Jan 13 '17 at 6:15

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.