The following is a theorem that I have some difficulty at it.
I do not know how the author shows that $\alpha \in \ell^1$. Please help me. Thanks in advance.
2026-03-29 19:09:17.1774811357
Norm of a sequence
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The property that the author is using is that, for a sequence $\alpha$, $$ \|\alpha\|_1=\sup\{\,|\langle \alpha,\beta\rangle|\,:\ \|\beta\|_\infty=1\,\}. $$ This is easily shown. Note that, by Hölder, if $\|\beta\|_\infty=1$, then $$ |\langle\alpha,\beta\rangle|\leq\|\alpha\|_1\,\|\beta\|_\infty=\|\alpha\|_1. $$ And $\|\alpha\|_1=\langle\alpha,\beta\rangle$ where $\beta_n$ is the complex number of absolute value one such that $\beta_n\alpha_n=|\alpha_n|$.