Littlewood's 4/3 inequality explained

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

c _{0} , the Banach space of scalar sequences that converge to zero.

Precisely, let

B:c _{0 x} c _{0} \to C or R be a bilinear form. Then the following holds:\left( \sum ^{inf ty} _{i,j =1} |B(e _{i,e} ^{4 / 3} ^{j)|} \right) ^{3/4} \le \sqrt{ 2 } \| B \|,

where

\| B \| = \sup \{|B(x _{1,x} _{2)|:} \|x _{i\|} _{infty} \le 1 \}.

The exponent 4/3 is optimal, i.e., cannot be improved by a smaller exponent.^[2] It is also known that for real scalars the aforementioned constant is sharp.^[3]

Generalizations

Bohnenblust–Hille inequality

Bohnenblust–Hille inequality^[4] is a multilinear extension of Littlewood's inequality that states that for all

m -linear mapping M : c _{0 x} … x c _{0} \to C the following holds:\left( \sum ^{inf ty} _{i _{1,\ldots,i} _{m=1}} |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

Notes and References

1. Littlewood. J. E.. On bounded bilinear forms in an infinite number of variables. The Quarterly Journal of Mathematics. 1930. 1. 164–174. os-1. 10.1093/qmath/os-1.1.164. 1930QJMat...1..164L.
2. Littlewood. J. E.. On bounded bilinear forms in an infinite number of variables. The Quarterly Journal of Mathematics. 1930. 1. 164–174. 10.1093/qmath/os-1.1.164. 1930QJMat...1..164L.
3. Diniz. D. E.. Munoz. G.. Pellegrino. D.. Seoane. J.. Lower bounds for the Bohnenblust--Hille inequalities: the case of real scalars. Proceedings of the American Mathematical Society. 2014. 132. 575–580. 10.1090/S0002-9939-2013-11791-0 . 1111.3253. 119128323 .
4. Bohnenblust. H. F.. Hille. Einar. On the Absolute Convergence of Dirichlet Series. The Annals of Mathematics. 1931. 32. 3. 600–622. 10.2307/1968255. 1968255 .