Is there a series that's unconditionally convergent but not absolutely convergent?

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- May 18th 2010, 02:52 PMchiph588@Unconditional Convergence
Is there a series that's unconditionally convergent but not absolutely convergent?

- May 18th 2010, 04:38 PMDrexel28
- May 18th 2010, 04:51 PMchiph588@
- May 18th 2010, 04:54 PMchiph588@
Wait a minute: "Absolute convergence and convergence together imply unconditional convergence, but unconditional convergence does not imply absolute convergence in general".

This is from Wikipiedia.

I'm really curious to see a series with this property. - May 18th 2010, 05:13 PMDrexel28
- May 19th 2010, 12:28 AMkompik
The two properties are equivalent in R and in finitely-dimensional spaces.

Try $\displaystyle X=\ell_2$ and the series $\displaystyle x_n=\frac 1{\sqrt{n}}e_n$, where $\displaystyle e_n$ denotes the sequence which has one on n'th place and all other terms are zeros.

If I remember correctly, this is precisely the most basic example given in the book

Kadets, Kadets: Series in Banach spaces: conditional and unconditional convergence

which is devoted basically to this topic.

However, many texts on functional analysis mention the relationship between various modes of convergence in Banach spaces.

I remember I have seen this in Wojtaszczyk's Banach spaces for analysts (with proofs) and in Megginson's Intorduction to Banach spaces theory (as an exercise - if I remember correctly).