Littlewood's 4/3 inequality

In mathematical analysis, Littlewood's 4/3 inequality, named after John Edensor Littlewood,^[1] is an inequality that holds for every complex-valued bilinear form defined on c₀, the Banach space of real sequences that converge to zero.

Precisely, let B:c₀ × c₀ → ℂ be a bilinear form. Then the following holds:

$\left( \sum_{i,j=1}^\infty |B(e_i,e_j)|^{4/3} \right)^{3/4} \le \sqrt{2} \| B \|,$

where

$\| B \| = \sup \{|B(x_1,x_2)|: \|x_i\|_\infty \le 1 \}.$

Generalizations

Bohnenblust–Hille inequality

Bohnenblust–Hille inequality^[2] is a multilinear extension of Littlewood's inequality that states that for all m-linear mapping M:c₀ × ... × c₀ → ℂ the following holds:

$\left( \sum_{i_1,\ldots,i_m=1}^\infty |M(e_{i_1},\ldots,e_{i_m})|^{2m/(m+1)} \right)^{(m+1)/(2m)} \le 2^{(m-1)/2} \| M \|,$

See also

• Grothendieck inequality

References

1. Littlewood, J. E. (1930). "On bounded bilinear forms in an infinite number of variables". The Quarterly Journal of Mathematics (1): 164–174. doi:10.1093/qmath/os-1.1.164.
2. Bohnenblust, H. F.; Hille, Einar (1931). "On the Absolute Convergence of Dirichlet Series". The Annals of Mathematics 32 (3): 600–622. doi:10.2307/1968255.