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 $B:c_0\times c_0\to ℂ$ or $ℝ$ be a bilinear form. Then the following holds:

${(\sum _{i,j=1}^{\mathrm{\infty }}|B(e_i,e_j){|}^{4/3})}^{3/4}\le \sqrt{2}\Vert B\Vert ,$

where

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

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 $M:c_0\times \cdots \times c_0\to ℂ$ the following holds:

${(\sum _{i_1,\dots ,i_m=1}^{\mathrm{\infty }}|M(e_{i_1},\dots ,e_{i_m}){|}^{2m/(m+1)})}^{(m+1)/(2m)}\le 2^{(m-1)/2}\Vert M\Vert ,$

See also

• Grothendieck inequality

References

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