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 ${\displaystyle c_{0}}$, the Banach space of scalar sequences that converge to zero.

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

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

where

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

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.