Timeline for Banach space with uncountable basis
Current License: CC BY-SA 4.0
13 events
| when toggle format | what | by | license | comment | |
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| Nov 26, 2022 at 11:59 | vote | accept | Anupam | ||
| Nov 20, 2022 at 16:52 | answer | added | Robert Furber | timeline score: 10 | |
| Nov 15, 2022 at 19:50 | comment | added | Michael Greinecker | @MartinSleziak Also A Banach space of (Hamel) dimension $\kappa$ exists if and only if $\kappa^{\aleph_0}=\kappa$, which answers the question in the title. | |
| Nov 15, 2022 at 19:37 | comment | added | Martin Sleziak | There is this question on Mathematics: Can any uncountable dimensional real vector space be made into a Banach space?. (Without a satisfactory answer, at the moment.) And there are some related questions about Hilbert spaces, as already pointed out: Can you equip every vector space with a Hilbert space structure? and Can all real/complex vector spaces be equipped with a Hilbert space structure? | |
| Nov 15, 2022 at 19:32 | history | edited | Martin Sleziak | CC BY-SA 4.0 |
edited body; edited tags
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| Nov 15, 2022 at 19:27 | comment | added | Christian Remling | @WillieWong: What I wrote was ambigious and you gave it a stronger than intended interpretation. I just wanted to point out that if there is a Banach space of a given algebraic dimension, then (trivially) any vector space of that dimension will do. | |
| Nov 15, 2022 at 18:24 | comment | added | Willie Wong | This MSE comment further suggests that there are non-trivial restrictions on the Hamel dimensions of Banach spaces (though I don't understand it at all). I do think this is a suitable question for MO. | |
| Nov 15, 2022 at 18:17 | comment | added | Willie Wong | This MSE answer gives a citation that every infinite dimensional separable Banach space has cardinality exactly $2^{\aleph_0}$. But is there any known result of limits on the cardinalities of non-separable Banach spaces? (Also @MartinBrandenburg.) | |
| Nov 15, 2022 at 18:12 | comment | added | Willie Wong | @ChristianRemling: is it obvious? The Theorem is that if $X$ is Banach, then its Hamel dimension is not $\aleph_0$. I think the OP is asking whether for any cardinal number other than $\aleph_0$ there is a Banach space with that Hamel dimension. | |
| Nov 15, 2022 at 18:06 | review | Close votes | |||
| Nov 21, 2022 at 15:55 | |||||
| Nov 15, 2022 at 17:54 | comment | added | Christian Remling | A vector space has no structure other than its dimension, so there can't be any obstructions. | |
| Nov 15, 2022 at 17:37 | comment | added | Gerald Edgar | You are asking in the wrong forum. When you go to math.meta.stackexchange.com/questions/9959 , do some searching to find your answer: For example, a Hamel basis for a Banach space cannot be countably infinite. | |
| Nov 15, 2022 at 17:33 | history | asked | Anupam | CC BY-SA 4.0 |