Grothendieck inequality
In mathematics, the Grothendieck inequality states that there is a universal constant k with the following property. If ai,j is an n by n (real or complex) matrix with
for all (real or complex) numbers si, tj of absolute value at most 1, then
- ,
for all vectors Si, Tj in the unit ball B(H) of a (real or complex) Hilbert space H. The smallest constant k which satisfies this property for all n by n matrices is called a Grothendieck constant and denoted k(n). In fact there are two Grothendieck constants kR(n) and kC(n) for each n depending on whether one works with real or complex numbers, respectively.[1]
The Grothendieck inequality and Grothendieck constants are named after Alexander Grothendieck, who proved the inequality and the existence of the constants in a paper published in 1953.[2]
Bounds on the constants
The sequences kR(n) and kC(n) are easily seen to be increasing, and Grothendieck's result states that they are bounded,[2][3] so they have limits.
With kR defined to be supn kR(n)[4] then Grothendieck proved that: .
Krivine (1979)[5] improved the result by proving: 1.67696... ≤ kR ≤ 1.7822139781...=, conjecturing that the upper bound is tight. However, this conjecture was disproved by Braverman et al. (2011).[6]
See also
References
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External links
- Weisstein, Eric W., "Grothendieck's Constant", MathWorld. (NB: the historical part is not exact there.)