Dual of the convergent sequence space ($c^{\prime }\cong {\ell }^1(\{\infty ,1,2,\dots \})$)

For the Banach space $c$ of convergent complex sequences with the supremum norm, every bounded complex-linear functional has the unique representationRestrict $F$ 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 $a_{\infty }=F(1)-{\sum }_j{a}_j$. For the norm lower bound choose the first $N$ coordinates of $x$ with phases maximizing $a_j{x}_j$ 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.