Littlewood's 4/3 inequality

In mathematical analysis, Littlewood's 4/3 inequality, named after John Edensor Littlewood, is an inequality that holds for every complex-valued bilinear form defined on $c_{0}$, the Banach space of scalar sequences that converge to zero.

Precisely, let $B:c_{0}\times c_{0}\to \mathbb {C}$ or $\mathbb {R}$ be a bilinear form. Then the following holds:

$\left(\sum _{i,j=1}^{\infty }|B(e_{i},e_{j})|^{4/3}\right)^{3/4}\leq {\sqrt {2}}\|B\|,$

where

$\|B\|=\sup\{|B(x_{1},x_{2})|:\|x_{i}\|_{\infty }\leq 1\}.$

The exponent 4/3 is optimal, i.e., cannot be improved by a smaller exponent. It is also known that for real scalars the aforementioned constant is sharp.

Generalizations

Bohnenblust–Hille inequality

Bohnenblust–Hille inequality is a multilinear extension of Littlewood's inequality that states that for all $m$-linear mapping $M:c_{0}\times \cdots \times c_{0}\to \mathbb {C}$ the following holds:

$\left(\sum _{i_{1},\ldots ,i_{m}=1}^{\infty }|M(e_{i_{1}},\ldots ,e_{i_{m}})|^{2m/(m+1)}\right)^{(m+1)/(2m)}\leq 2^{(m-1)/2}\|M\|,$

See also

• Grothendieck inequality