Timeline for What well known results with countability assumptions can be naturally extended to uncountable settings?
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6 events
| when toggle format | what | by | license | comment | |
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| Apr 8, 2022 at 9:11 | comment | added | Jochen Wengenroth | The equivalence of weak compactness and weak sequential compactness is true for all normed spaces (and also for all metrizable locally convex spaces). This is (a version of) the Eberlen-Smulian theorem. | |
| Jan 7, 2022 at 14:31 | comment | added | Hua Wang | I learned this as a corollary of two basic theorems in Banach space theory: one is the characterization (due to Goldstine?) of reflexive Banach space as the ones in which the closed unit ball is weakly compact, another is the Eberlein-Smulian theorem. I know this is not a geodesic approach, but maybe many learned this fact in a similar way as I did, in which case, one naturally arrives at this conclusion without the separability assumption. | |
| Jan 7, 2022 at 14:21 | comment | added | coudy | "People think" that separability is needed because they learn the Hilbert space case as a corollary of the Banach space case. There are no closed supplementary subspace in general for closed subspaces of Banach spaces and thus the last part of the argument fails. Hence the need for Tykhonov and AC to get "open cover" compactness of the unit ball (of the dual). Separability gives the metrizability of the unit ball for the weak topology and finally its sequential compactness. | |
| Jan 7, 2022 at 11:45 | history | made wiki | Post Made Community Wiki by Stefan Kohl♦ | ||
| Jan 7, 2022 at 9:42 | comment | added | YCor | But (as you say) being sequentially compact is quite clearly a property that holds for a space whenever it holds for closure of countable subsets... I'm quite puzzled by the claim "people often think that separability is needed here"... | |
| Jan 6, 2022 at 21:06 | history | answered | coudy | CC BY-SA 4.0 |