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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.

Tomasz Kania
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CSH
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1 Answers1

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

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Tomasz Kania
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  • 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. – Aweygan Jan 12 '17 at 20:46
  • How can I relate sequentially closed and sequentially compact? Are they equivalent? – CSH Jan 13 '17 at 00:02
  • @Aweygan, there are two questions: the first one is about metrasibility/sequentiality of $X^$ in the weak-topology. – Tomasz Kania Jan 13 '17 at 06:15