1
$\begingroup$

Let $X$ be a Banach space, and $P$ be a projection of $X^*$ such that $\Vert P \Vert\leq 1$ and $\ker(P)$, ${\bf ran}(1-p)$ are weak-star closed. Show that $P$ is weak-star to weak-star continuous.

I welcome if any body give hints or any ideas to prove it.

Improve this question
$\endgroup$
5
  • 3
    $\begingroup$ Why the soft-question tag? Is this an exercise? $\endgroup$
    – Jochen Wengenroth
    Commented Feb 10, 2018 at 9:11
  • $\begingroup$ @ Jochen, I feel it should be simple proof which I am not getting. $\endgroup$
    – Andy
    Commented Feb 10, 2018 at 10:08
  • 2
    $\begingroup$ $P$ is clearly weak$^*$ continuous on bounded sets. If you want to avoid using the full force of the Krein-Smulian Theorem, use the simple lemma Robert Whitlley proved in ams.org/journals/proc/1986-097-02/S0002-9939-1986-0835903-X/… to show that $P^*$ maps $X$ back into $X$. $\endgroup$
    – Bill Johnson
    Commented Feb 10, 2018 at 18:05
  • 1
    $\begingroup$ R. Whitley, The Krein-Smulian Theorem, PAMS 97 (1986), 376-377. $\endgroup$
    – Bill Johnson
    Commented Feb 10, 2018 at 18:08
  • $\begingroup$ The "soft-question" tag is not meant for easy questions, but for questions that do not necessarily have precise interpretations or precise answers $\endgroup$
    – Yemon Choi
    Commented Feb 11, 2018 at 3:17

0

Reset to default

Browse other questions tagged .