2
$\begingroup$

Let $X$ be a normed space such that for all $(a_n)_{n=1}^{\infty}\subset X$ satisfying $\sum_{n=1}^{\infty}\|a_n\|<\infty$, it follows that $\sum_{n=1}^{\infty}a_n$ converges to a limit in $X$. Show that $X$ is a Banach space.

I had been asked that question (to be precise, I have been asked to show "if and only if") whereas all I managed to prove is the other direction. I sincerely have got no idea how to compare a sequence $(x_n)_{n=1}^{\infty}\subset X$ that is Cauchy to a series that converges absolutely. Could you give me a hint? I have been looking for such a question and come by nothing, following which, I suspect, the statement is either wrong or alternately immediate and simple.

  • 1
    I just came by one. Many times it is only after I post a question that I actually find a duplicate, unfortunately – 2017-02-26
  • 0
    To write a norm use "\|" instead of "||". – 2017-02-26
  • 3
    Hint: Take a Cauchy sequence. Then use the Cauchy criterion to choose a subsequence where adjacent elements in the sequence are very close to each other. Then form a telescoping sum (sum of the differences between the adjacent elements) – 2017-02-26

0 Answers 0