Answer to Question #139801 in Functional Analysis for anjali

Question #139801
In a Banach space, show that every absolutely convergent series is convergent.
1
Expert's answer
2020-10-25T18:36:04-0400

Let "X" be Banach space.

Let "\\sum\\limits_{k=1}^{\\infty}a_k" be an absolutely convergent series in "X" , that is "\\sum\\limits_{k=1}^{\\infty}\\|a_k\\|" is a convergent series.

Denote "\\sum\\limits_{k=1}^na_k" by "S_n", and let "S_0=0".

By the Cauchy's convergence test we have that for every "\\varepsilon>0" there is "N\\in\\mathbb N" such that "\\sum\\limits_{k=n}^m\\|a_k\\|<\\varepsilon" for every "n,m>N, m\\ge n".

By the general triangle inequality we have "\\|S_m-S_{n-1}\\|=\\bigl\\|\\sum\\limits_{k=n}^ma_k\\bigr\\|\\le\\sum\\limits_{k=n}^m\\|a_k\\|", so "\\{S_n\\}_{n\\in\\mathbb N}" is a fundamental sequence in "X". Since "X" is Banach space, we have that "\\{S_n\\}_{n\\in\\mathbb N}" is a convergent sequence, that is "\\sum\\limits_{k=1}^{\\infty}a_k" is a convergent series.


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