Timeline for Dual space of $\ell^\infty$
Current License: CC BY-SA 4.0
8 events
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| Nov 1 at 17:53 | history | edited | Pietro Majer | CC BY-SA 4.0 |
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| Nov 13, 2019 at 21:15 | comment | added | user131781 | Just a wee remark. One can avoid the use of the double dual of $B(H)$ in this argument by employing one of the vector-valued versions of the Riesz representation theorem to show that this operator is integration with respect to a measure with values in $B(H)$—-this works because the unit ball of the latter is compact for a suitable topology, the weak operator topology. | |
| Jan 25, 2018 at 19:36 | history | edited | Pietro Majer | CC BY-SA 3.0 |
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| Jan 24, 2018 at 23:34 | comment | added | Pietro Majer | @NikWeaver: Wonderful | |
| Jan 24, 2018 at 23:26 | comment | added | Nik Weaver | @YemonChoi You mean, to get multiplicativity? Sure, I guess you can do it that way. Probably I would just make a continuity argument, given that the continuous functional calculus is multiplicative. | |
| Jan 24, 2018 at 22:56 | comment | added | Yemon Choi | @NikWeaver Aren't you using Arens products somewhere in the background, then? | |
| Jan 24, 2018 at 22:55 | comment | added | Nik Weaver | I use this fact to give a quick and easy construction of the Borel functional calculus for a self-adjoint operator $A \in B(H)$. The continuous functional calculus is an isometric linear map from $C({\rm spec}(A))$ into $B(H)$. Double dualizing, we get an isometric linear map from $C({\rm spec}(A))^{**}$ into $B(H)^{**}$. Since $B(H)$ is a dual space, we can project that $B(H)^{**}$ onto $B(H)$; restricting the domain to ${\rm Bor}({\rm spec}(A)) \subseteq C({\rm spec}(A))^{**}$ then yields the Borel functional calculus. | |
| Jan 24, 2018 at 21:52 | history | answered | Pietro Majer | CC BY-SA 3.0 |