It is well known that in Banach spaces absolute convergence implies convergence. Since this is a characterization of Banach spaces, it does not hold in others kinds of spaces. In particular, let we consider the normed vector space $( C_c(\mathbb R), \lVert \cdot \rVert_{C^0} )$ of continuous functions which support is compact, endowed with the maximum norm ( $\lVert f \rVert_{C^0} := \max{ \{ \lvert f(x) \rvert : x \in \mathbb R \}}$ ). This space is not complete, so it is not a Banach space and hence must exist series that are absolutely convergent but not convergent. Sadly, I can't find them. May anyone help me?
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I think this works : Just take a function $f_n$ supported on $[n,n+1]$ such that $\|f_n\| = 1/n^2$. Then it is absolutely convergent, but if $$ f = \sum f_n $$ Then $f$ cannot be compactly supported.
answered Aug 20 '13 at 18:10
