Coordinate functionals on sequence spaces

Given: The spaces, maps, scalar field, and hypotheses in the statement above. All duals consist of linear functionals over the ambient field; evaluation has no conjugation.

From The continuous dual of c0 is ell-one, with its stated hypotheses: Let $\mathbb{K}=\mathbb{R}$ or $\mathbb{C}$. With coordinates starting at zero, the map ${\mathrm{\ell }}^1(\mathbb{K})⟶c_0(\mathbb{K}{)}^{\ast },a⟼f_a,f_a(x)={\sum }_{n=0}^{\mathrm{\infty }}{a}_n{x}_n$ is a linear isometric bijection. The pairing is bilinear, including over $\mathbb{C}$.

From The complex continuous dual of ell-one is ell-infinity, with its stated hypotheses: For complex sequence spaces, with indices starting at zero, ${\mathrm{\ell }}^{\mathrm{\infty }}(\mathbb{C})⟶{\mathrm{\ell }}^1(\mathbb{C}{)}^{\ast },b⟼h_b,h_b(a)={\sum }_{n=0}^{\mathrm{\infty }}{b}_n{a}_n$ is a complex-linear isometric bijection. There is no conjugation in this pairing.

From Counting measure specializes the representation theorem to ${\mathrm{\ell }}^p$ and ${\mathrm{\ell }}^q$, with its stated hypotheses: Let $1\le p<\mathrm{\infty }$ and let $q$ be conjugate to $p$. Every bounded linear functional $\mathrm{\Lambda }:{\mathrm{\ell }}^p\to \mathbb{R}$ is of the form $\mathrm{\Lambda }(a)={\sum }_{n=0}^{\mathrm{\infty }}{a}_n{b}_n$ for a unique sequence $b\in {\mathrm{\ell }}^q$. Moreover, $\mathrm{\parallel }\mathrm{\Lambda }\mathrm{\parallel }=\mathrm{\parallel }b{\mathrm{\parallel }}_{{\mathrm{\ell }}^q}\mathrm{.}$