= Absolutely summable sequence space
{title2=$\ell^1=\{x:\sum_n|x_n|<\infty\}$}
= l1 sequence space
{c}
{synonym}
The <Banach space> $\ell^1$ consists of absolutely summable scalar sequences, with <norm> $\|x\|_1=\sum_n|x_n|$. Its continuous dual is $\ell^\infty$, under the pairing $\sum_nx_ny_n$. Finitely supported sequences are <norm> dense, which identifies a <linear functional> from its values on coordinate vectors. The <space of sequences converging to zero> inside this dual is a <norming subspace> of infinite codimension.
For example, $x=(2^{-1},2^{-2},\ldots)$ belongs to the <absolutely summable sequence space> and has <norm> one, by the <geometric series> formula. Merely tending to zero is insufficient: $(1/n)$ tends to zero but its <series> of absolute values diverges. Thus this <sequence space> records total absolute size, rather than just the eventual size of individual coordinates. Its <completeness> ensures that a <Cauchy sequence> in this total-error <norm> has an absolutely summable limit.https://people.clas.ufl.edu/mjury/files/6617notes30nov2015.pdf[Michael Jury, MAA6617 course notes, section 20.1(b): Banach sequence spaces]{ref}
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