Cotype 2 ($C_2(X)$)

This is cotype of a Banach space with $q=2$. The Lp space $L^1$ has cotype 2 with constant at most $\sqrt{2}$: the triangle inequality for the Euclidean norm gives $(\sum \Vert f_j{\Vert }_1^2{)}^{1/2}\le \int (\sum |f_j(t){|}^2{)}^{1/2}dt$, and the sharp Rademacher second-moment inequality, followed by Fubini theorem, bounds this by $\sqrt{2}\mathbb{E}\Vert \sum {\epsilon }_j{f}_j{\Vert }_1$.

If a Banach space $Y$ has cotype 2 constant $C$, every bounded linear operator $T:{\ell }_{\infty }^N\to Y$ satisfies ${\pi }_2(T)\le 3^{1/4}C{\pi }_4(T)$ and ${\pi }_2(T)\le 4\sqrt{3}{C}^2\Vert T\Vert$, uniformly in $N$. The first bound follows from Pietsch factorization theorem and the fourth moment bound for Rademacher sums. For the second, 2-summing domination gives a weighted coordinate $L^2$ norm; clipping coordinates at a variable threshold yields ${\pi }_4(T)\le 2\sqrt{\Vert T\Vert {\pi }_2(T)}$, which absorbs the first bound.