The James formula defines a norm

For fixed $p$, $q_p(x)$ is $2^{-1\mathrm{/}2}$ times the Euclidean norm of the finite vector of cyclic successive differences. Euclidean Minkowski gives $q_p(x+y)\le q_p(x)+q_p(y)$ and homogeneity is immediate. Taking suprema shows that $J$ is a vector subspace and gives the triangle inequality and homogeneity for $\mathrm{\parallel }\cdot {\mathrm{\parallel }}_J$. [L1, Euclidean Minkowski] (Euclidean Minkowski).

For $i<j$, the two-point tuple $(i,j)$ gives $q_{(i,j)}(x)=\mathrm{\mid }{x}_i-x_j\mathrm{\mid }$. Since $x_j\to 0$ for $x\in c_0$, letting $j\to \mathrm{\infty }$ yields $\mathrm{\mid }{x}_i\mathrm{\mid }\le \mathrm{\parallel }x{\mathrm{\parallel }}_J$. Taking the supremum proves the first displayed inequality and definiteness, including at $x=0$. [L1, two-point tuple] (two-point tuple).

For $x\in {\mathrm{\ell }}^2$ and $p=(p_1<\cdots <p_k)$, use $\mathrm{\mid }a-b{\mathrm{\mid }}^2\le 2\mathrm{\mid }a{\mathrm{\mid }}^2+2\mathrm{\mid }b{\mathrm{\mid }}^2$ in the cyclic sum. Every selected coordinate occurs twice before the factor $1\mathrm{/}2$, so $q_p(x{)}^2\le 2{\sum }_{j=1}^k\mathrm{\mid }{x}_{p_j}{\mathrm{\mid }}^2\le 2\mathrm{\parallel }x{\mathrm{\parallel }}_2^2$. Taking the supremum gives ${\mathrm{\ell }}^2\subseteq J$ and the second inequality. [L1, finite estimate] (finite estimate).