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I have problems with this question:

Denote $E$ the set of all real sequences $\{a_n\}$ such that $|a_n|\le 1$ for every positive integers $n$, $l^1$ be the set of all real sequences $\{a_n\}$ such that $ \sum a_n$ converges absolutely, $l^\infty$ be the set of all real sequences $\{a_n\}$ such that $\{a_n\}$ is bounded, and $c_0$ be the set of all real sequences $\{a_n\}$ such that $\lim_{n\to\infty}a_n=0$.

Let $\{a_n\}$ be a sequence such that $\{a_nb_n\}\in l^1$ for every sequence $\{b_n\}\in l^1$. Prove that $\{a_n\}\in l^\infty$. Show that this statement is false if $l^\infty$ is replaced by $c_0$.

Any help?

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1 Answer 1

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Suppose that the sequence $(a_n ) $ is not bounded. Then there exists a sequence $(n_k)$ of natural numbers such that $$|a_{n_k} |\geq 2^k.$$ Define sequence $(b_n) $ as follows $$b_n =\begin{cases} 0 \hspace{0.5cm}\mbox{ if } n\in\mathbb{N}\setminus \bigcup_{k=1}^{\infty} \{n_k \} \\ \frac{a_{n_k}}{a_{n_k}^2} \mbox{ if } n=n_k \end{cases}.$$ Clearly $(b_n )\in \ell^1$ but $(a_n b_n )\notin \ell^1. $ Contradiction.

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