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