Convergent subsequence under image of adjoint.

Asked Mar 09 '22 at 22:48

Active Mar 10 '22 at 12:48

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Let $X$ and $Y$ be normed spaces, and let $T : X \rightarrow Y $ be a bounded linear operator, and denote by $T^*$ the adjoint operator of $T$.

Suppose also that $X$ is separable, that is, $X$ admits a countable dense subset. Let $ (y_n)_{n \in \mathbb{N} } $ be a sequence of bounded linear functionals in $Y^*$. Prove that if $(y_n)_{n \in \mathbb{N} }$ is bounded, then $ T^*(y_n)_{n \in \mathbb{N}} $ has a convergent subsequence.

Quite stuck on this, any hints or tips?

edited Mar 10 '22 at 12:48

asked Mar 09 '22 at 22:48

turnip_man

You're correct, my mistake. – turnip_man Mar 10 '22 at 00:21
What is $g_n$ and what have you tried? – Kavi Rama Murthy Mar 10 '22 at 04:50
https://en.wikipedia.org/wiki/Banach%E2%80%93Alaoglu_theorem – daw Mar 10 '22 at 07:42
As stated, this is wrong but $T^(g_n)$ has a weak$^$-convergent subsequence (use daw's hint to Alaoglu and metrizability of the dual unit ball of $X$ in the weak$^*$-topology). – Jochen Mar 10 '22 at 08:29

Convergent subsequence under image of adjoint.

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