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}\leq {\sqrt {2}}\|B\|,

where

\|B\|=\sup\{|B(x_{1},x_{2})|:\|x_{i}\|_{\infty }\leq 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)}\leq 2^{(m-1)/2}\|M\|,

References

↑ 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.
↑ 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.

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Littlewood's 4/3 inequality

Generalizations

Bohnenblust–Hille inequality

See also

References