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How to prove $\sum\limits_{k=1}^{\infty}|\alpha_{k}|\lt \infty$, given that $\sum \limits_{k=1}^{\infty}\alpha_{k}\phi_{k}$ converges …?
Below is a question that I came across while studying real analysis. Given that $\{a_k\}_{k=1}^{\infty}$ is a sequence of complex numbers where $\sum_k{a_k}{b_k}$ converges for every complex sequence $\{b_k\}_{k=1}^{\infty}$ such that $\lim_kb_k=0$. Prove that $\sum_k|{a_k}|<\infty$. As a matter of fact as I am very new to this field. I need some help. Thanks.