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1. The problem statement, all variables and given/known data

Question:

Let L be a linear functional/form on a real Banach space X and let {x_k} be a sequence of vectors such that L(x_k) converges. Can I conclude that {x_k} has a limit in X?

It would help me greatly in solving a certain problem if I knew the answer to that question.

3. The attempt at a solution

The natural approach is to try to show that {x_k} is Cauchy.

Since the sequence of real numbers {L(x_k)} converges, then it is Cauchy, so for n,k large enough,

[tex]|L(x_k)-L(x_n)|=|L(x_k - x_n)|<\epsilon[/tex]

Now what??

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# Linear forms and complete metric space

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