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For isomorphic finite-dimensional normed vector spaces, the Banach-Mazur distance is the infimum of $\Vert S\Vert \Vert S^{-1}\Vert$ over their linear isomorphisms. If an $N$-dimensional subspace $E$ of a Banach space of cotype 2 satisfies the summing estimate ${\pi }_2(T)\le L\Vert T\Vert$ for operators from ${\ell }_{\infty }^N$, then $d(E,{\ell }_{\infty }^N)\ge \sqrt{N}/L$: compose an isomorphism with the inclusion and use ${\pi }_2(I_E)=\sqrt{N}$.

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.

We use independent symmetric Rademacher random variables ${\epsilon }_j\in \{-1,1\}$, the intended meaning of the Bernoulli variables in this context. Without this convention the assertion need not hold: for one $0/1$ Bernoulli variable of success probability $t<1/2$ and one nonzero vector, the proposed inequality would be $\sqrt{t}\le \sqrt{2}t$, which is false.

Let $\Omega =\{-1,1{\}}^n$ with the uniform probability measure, put $S(\epsilon )={\sum }_j{\epsilon }_j{x}_j$, and set $F(\epsilon )=\Vert S(\epsilon )\Vert$. We prove the sharp estimate rather than assuming the Kahane-Khintchine inequality. If ${\epsilon }^{(j)}$ denotes $\epsilon$ with its $j$th sign reversed, defineThe Walsh characters $w_A(\epsilon )={\prod }_{j\in A}{\epsilon }_j$ form an orthonormal basis of scalar functions on this finite cube, and $\mathcal{L}{w}_A=-|A|w_A$. Since $F(-\epsilon )=F(\epsilon )$, its coefficients for odd $|A|$ vanish. Every nonconstant remaining character has $|A|\ge 2$. Expanding in this orthonormal basis gives the even-function spectral gap on a hypercubeAt each $\epsilon$, the Hahn-Banach theorem supplies a real norming linear functional $\varphi$ with $\Vert \varphi \Vert \le 1$ and $\varphi (S(\epsilon ))=F(\epsilon )$. For a complex Banach space use its underlying real space. Since $\Vert z\Vert \ge \varphi (z)$,Adding gives $\mathcal{L}F\ge -F$. Multiplying by the nonnegative $F$ yields $F(-\mathcal{L}F)\le F^2$. ThereforeThe constant is sharp: take $n=2$ and $x_1=x_2\ne 0$ in the scalar field. The random norm is $0$ or $2\Vert x_1\Vert$, each with probability one half.

A Banach space has cotype 2 if a constant $C$, independent of the length of the family, satisfiesFor $f_j\in L^1(0,1)$, the triangle inequality for the Euclidean norm givesFor real or complex scalars $a_j$, independence gives $\mathbb{E}|{\sum }_j{\epsilon }_j{a}_j{|}^2={\sum }_j|a_j{|}^2$. The proved sharp Rademacher second-moment inequality, applied pointwise to $(f_j(t))$, and Fubini theorem now yieldThus $L^1(0,1)$ has cotype 2, with $C\le \sqrt{2}$.

Let $C$ be a cotype 2 constant of $Y$. All bounded linear operators from ${\ell }_{\infty }^N$ have finite p-summing norms: writing $Tx={\sum }_{k=1}^N{x}_k{y}_k$ and using the triangle inequality in the finite-family ${\ell }^p$ norm gives ${\pi }_p(T)\le {\sum }_k\Vert y_k\Vert$. Thus applying the Pietsch factorization theorem below is legitimate.

First let $a_4={\pi }_4(T)$, and use the Pietsch factorization theorem to choose a probability measure $\mu$ on the dual unit ball $B_{{\ell }_1^N}$. For a finite family $x_j\in {\ell }_{\infty }^N$, the cotype 2 inequality and Pietsch factorization theorem domination giveThe Jensen inequality bounds the final bracket by $[\int \mathbb{E}|{\sum }_j{\epsilon }_j\varphi (x_j){|}^{4}d\mu {]}^{1/4}$. For real or complex scalars,For complex scalars, paired-index expansion gives $\mathbb{E}|{\sum }_j{\epsilon }_j{b}_j{|}^4=2({\sum }_j|b_j{|}^2{)}^2+|{\sum }_j{b}_j^2{|}^2-2{\sum }_j|b_j{|}^4$, which is at most the displayed bound by the triangle inequality. For real scalars this reduces to $3(\sum b_j^2{)}^2-2\sum b_j^4$. Since $\mu$ is a probability measure on the dual unit ball, we obtainTaking the least constant proves ${\pi }_2(T)\le K{\pi }_4(T)$ with $K=3^{1/4}C$, uniformly in $N$.

We next prove a summing-norm interpolation estimate using coordinate truncation. Put $a={\pi }_2(T)$ and $M=\Vert T\Vert$, assuming $T\ne 0$. The Pietsch factorization theorem for $p=2$ gives a measure $\mu$ on $B_{{\ell }_1^N}$. The Cauchy-Schwarz inequality givesSet ${\nu }_i=\int |{\varphi }_i|d\mu$. Their sum is at most one; adding the missing mass to one coordinate, if necessary, makes $({\nu }_i)$ a probability vector and preserves domination. HenceWrite $r=({\sum }_i{\nu }_i|x_i{|}^4{)}^{1/4}$. For $t>0$, clip each coordinate of $x$ at magnitude $t$, obtaining $z_i=x_i\min (1,t/|x_i|)$, with $z_i=0$ if $x_i=0$. Set $w=x-z$. Then $\Vert z{\Vert }_{\infty }\le t$ andThe triangle inequality and the two different bounds on $T$ now give $\Vert Tx\Vert \le Mt+a{r}^2/t$. For $r>0$, take $t=r\sqrt{a/M}$; if $r=0$, domination already gives $Tx=0$. ConsequentlySumming fourth powers over any family $(x_j)$ givessince each coordinate evaluation belongs to the dual unit ball. Therefore ${\pi }_4(T)\le 2\sqrt{M{\pi }_2(T)}$. Combine this with the first estimate:The zero operator also satisfies this inequality. Both constants depend only on $Y$, through its cotype 2 constant, and not on $N$.

Finally let $E\subset Y$ have dimension $N$, with its inherited norm. The permitted finite-dimensional identity result is ${\pi }_2(I_E)=\sqrt{N}$. The inclusion $J:E\to Y$ is an isometric embedding, so ${\pi }_2(J)={\pi }_2(I_E)$. For every Banach space isomorphism $S:{\ell }_{\infty }^N\to E$, apply the preceding estimate to $T=JS$ and the ideal property of absolutely p-summing operators:The Banach-Mazur distance is the infimum of this last product over $S$, and therefore