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If $a,b \in \ell^2 $ then $a\bar b \in \ell^1$

($\ell^1$ and $\ell^2$ on $\mathbb C$ and $\bar b$ is complex conjugate of $b$)

What did I write :

Since $a,b \in \ell^2 $, $a=(a_n)$ and $b=(b_n)$ such that $\sum_{n=1}^{\infty}|a_n|^2 \lt \infty$ and $\sum_{n=1}^{\infty}|b_n|^2 \lt \infty$

But I cannot realize that how can I show $\sum_{n=1}^{\infty}|a_n\bar {b_n}| \lt \infty$ from above?

Thanks in advance

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Hint: apply the Cauchy-Schwartz inequality.

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  • $\begingroup$ Thank you. Also I wonder how can I show it same thing for Lebesgue spaces. Is Cauchy-Schwarz inequality still valid for integrals? $\endgroup$
    – user519955
    Oct 12 '18 at 20:56
  • $\begingroup$ @user519955 Yes it is $\endgroup$
    – JavaMan
    Oct 13 '18 at 2:45
  • $\begingroup$ thanks a lot :) $\endgroup$
    – user519955
    Oct 13 '18 at 9:36

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