Timeline for Algebraic Dual / Continuous Dual
Current License: CC BY-SA 2.5
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Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
replaced http://mathoverflow.net/ with https://mathoverflow.net/
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Feb 4, 2010 at 8:30 | comment | added | Yemon Choi | FWIW not every Banach space has a Schauder basis. If memory serves rightly, the first counter-example was due to Per Enflo. That said, the gap between Schauder basis and Hamel basis is vast in infinite dimensions | |
Feb 4, 2010 at 3:36 | comment | added | Jonas Meyer | Oh, in case it is confusing, the examples Ady gave are now in the comment by Scott Morrison, $l^\infty(\mathbb{N})$ and $C[0,1]^*$. | |
Feb 4, 2010 at 3:20 | comment | added | Jonas Meyer | (1)What about the examples Ady gives in the comments above? (2)What in general do you mean by topological basis? For Hilbert space I assume you mean orthonormal basis, and more generally some Banach spaces have Schauder bases, but I don't know if an analogous set exists in every Banach space. (3) Your characterization of the topological dual isn't accurate; you can't send the "topological basis" just anywhere. E.g., consider the identification of the dual of $l^2$ with $l^2$. (This point, however, only shows that the topological dual is "smaller" than you indicated.) | |
Feb 4, 2010 at 1:52 | history | answered | Ilya Grigoriev | CC BY-SA 2.5 |