Editing Absolute convergence (section)

===Proof that any absolutely convergent series in a Banach space is convergent===

The above result can be easily generalized to every [[Banach space]] <math>(X, \|\,\cdot\,\|).</math>  Let <math display="inline">\sum x_n</math> be an absolutely convergent series in <math>X.</math> As <math display=inline>\sum_{k=1}^n\|x_k\|</math> is a [[Cauchy sequence]] of real numbers, for any <math>\varepsilon > 0</math> and large enough [[natural number]]s <math>m > n</math> it holds:
<math display=block>\left| \sum_{k=1}^m \|x_k\| - \sum_{k=1}^n \|x_k\| \right| = \sum_{k=n+1}^m \|x_k\| < \varepsilon.</math>

By the triangle inequality for the norm {{math|ǁ⋅ǁ}}, one immediately gets:
<math display=block>\left\|\sum_{k=1}^m x_k - \sum_{k=1}^n x_k\right\| = \left\|\sum_{k=n+1}^m x_k\right\| \leq \sum_{k=n+1}^m \|x_k\| < \varepsilon,</math>
which means that <math display=inline>\sum_{k=1}^n x_k</math> is a Cauchy sequence in <math>X,</math> hence the series is convergent in <math>X.</math><ref>{{citation
|last = Megginson|first = Robert E.|author-link = Robert Megginson
|title = An introduction to Banach space theory
|series = Graduate Texts in Mathematics
|volume = 183 
|publisher = Springer-Verlag
|location = New York 
|year = 1998
|isbn = 0-387-98431-3
|page = 20
}} (Theorem 1.3.9)</ref>