Timeline for Non-linear weak*-continuous left inverses
Current License: CC BY-SA 4.0
6 events
| when toggle format | what | by | license | comment | |
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| Oct 5, 2020 at 13:28 | comment | added | Jochen Wengenroth | I have overlooked the "bi" of biduals. Just for duals of Banach spaces a counterexample is the restriction $\ell_\infty' \to c_0'$ which does not have a weak$^*$-continuous right inverse. | |
| Oct 5, 2020 at 10:15 | comment | added | Jochen Wengenroth | I doubt that there are always weak$^*$-continuous right inverses even for duals of Banach spaces. Assume that $X$ is a closed subspace of a Banach space $Y$ and assume that the quotient (restriction) map $q:Y'\to X'$ has a weak$^*$ continuous right inverse $r$. For every sequence $f_n$ of functionals on $X$ such that $f_n(x)\to 0$ pointwise on $X$ the sequence $r(f_n)$ would be a pointwise convergent sequence of extensions to $Y$ -- I don't believe that this is always possible and I am quite optimistic that the Banach spacers on MO can provide counterexamples. | |
| Oct 5, 2020 at 6:41 | comment | added | Jochen Wengenroth | I think you mean right inverses which always exist for quotient mappings between Fréchet spaces. The trivial example of the $0$ mapping $T:\mathbb R \to \{0\}$ shows that in many cases there are no left inverses. | |
| Oct 4, 2020 at 20:29 | history | edited | T. Milva | CC BY-SA 4.0 |
added 21 characters in body
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| Oct 4, 2020 at 8:27 | review | First posts | |||
| Oct 4, 2020 at 9:17 | |||||
| Oct 4, 2020 at 8:23 | history | asked | T. Milva | CC BY-SA 4.0 |