= Dual of the convergent sequence space
{title2=$c'\cong\ell^1(\{\infty,1,2,\ldots\})$}
For the <Banach space> $c$ of convergent complex sequences with the <supremum norm>, every bounded complex-<linear functional> has the unique representation
$$
F(x)=a_\infty\lim_jx_j+\sum_{j\geq1}a_jx_j,\qquad \|F\|=|a_\infty|+\sum_j|a_j|.
$$
Restrict $F$ to $c_0$ and use coordinate <duality of sequence spaces> to obtain $(a_j)\in\ell^1$. Splitting $x$ into its constant limit sequence and a zero-limit remainder forces $a_\infty=F(\mathbf1)-\sum_ja_j$. For the <norm> lower bound choose the first $N$ coordinates of $x$ with phases maximizing $a_jx_j$ and make its tail the constant phase maximizing $a_\infty\lim x$. The finite prefix and limit contributions attain their absolute values; the remaining contribution is bounded by the coefficient tail, which tends to zero. The opposite <norm> inequality is the <triangle inequality>. The index at infinity is an independent coordinate of the dual <norm>, rather than part of the sum of the finite-coordinate coefficients.
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