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 {\displaystyle c_{0}} , the Banach space of scalar sequences that converge to zero. Precisely, let B : c 0 × c 0 → C {\displaystyle B:c_{0}\times c_{0}\to \mathbb {C} } or R {\displaystyle \mathbb {R} }
be a bilinear form. Then the following holds:
( ∑ i , j = 1 ∞ | B ( e i , e j ) | 4 / 3 ) 3 / 4 ≤ 2 ‖ B ‖ , {\displaystyle \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 ‖ ∞ ≤ 1 } . {\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.
Generalizations
Bohnenblust–Hille inequality Bohnenblust–Hille inequality is a multilinear extension of Littlewood's inequality that states that for all m {\displaystyle m} -linear mapping
M : c 0 × ⋯ × c 0 → C {\displaystyle M:c_{0}\times \cdots \times c_{0}\to \mathbb {C} } the following holds:
( ∑ i 1 , … , i m = 1 ∞ | M ( e i 1 , … , e i m ) | 2 m / ( m + 1 ) ) ( m + 1 ) / ( 2 m ) ≤ 2 ( m − 1 ) / 2 ‖ M ‖ , {\displaystyle \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\|,}
See also Grothendieck inequality
References