James space

Let $c_0=c_0(\mathbb{N};\mathbb{R})$ as in The sequence spaces c_0 and ell-infinity, and use positive labels for its coordinates: if $\tilde{x}:\mathbb{N}\to \mathbb{R}$ is the underlying sequence, then $x_n$ below means ${\tilde{x}}_{n-1}$ for $n\ge 1$. Likewise $e_n$ denotes the canonical vector supported at the underlying coordinate $n-1$. If $p=(1\le p_1<\cdots <p_k)$ is a nonempty finite increasing tuple, define $q_p(x)=0$ for $k=1$, and for $k\ge 2$ define

$q_p(x{)}^2:=\frac{1}{2}({\sum }_{j=1}^{k-1}\mathrm{\mid }{x}_{p_j}-x_{p_{j+1}}{\mathrm{\mid }}^2+\mathrm{\mid }{x}_{p_k}-x_{p_1}{\mathrm{\mid }}^2)\mathrm{.}$

The James space is the real vector space

$J:=\{x\in c_0:{\sup }_p{q}_p(x)<\mathrm{\infty }\},$

equipped with $\mathrm{\parallel }x{\mathrm{\parallel }}_J:={\sup }_p{q}_p(x)$. The cyclic closing term and the factor $1\mathrm{/}2$ are part of this fixed norm; later equivalent James norms are not being identified with it isometrically.