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I'm stuck on a problem in functional analysis. Let x be a sequence on the Natural nummers such that for any square summable sequence y, the product sequence xy is absolutely summable. Then x is square summable.

Hint : Use the Closed graph theorem.

If I can prove the map Tx : y -> xy had a closed graph then It Follows from the Closed graph theorem that Tx is bounded and therefore that x is square summable, but I cant seem to show that the graph is Closed. Am I following the right path? Any hints?

Thanks