# 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

$\left| \sum_{i,j} a_{ij} s_i t_j \right|\le 1$

for all (real or complex) numbers si, tj of absolute value at most 1, then

$\left| \sum_{i,j} a_{ij} \langle S_i , T_j \rangle \right|\le k$,

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: $1.57 \approx \frac{\pi}{2} \leq k_{\R} \leq \mathrm{sinh}(\frac{\pi}{2}) \approx 2.3$.

Krivine (1979)[5] improved the result by proving: 1.67696... ≤ kR ≤ 1.7822139781...=$\frac{\pi}{2 \ln(1+\sqrt{2})}$, conjecturing that the upper bound is tight. However, this conjecture was disproved by Braverman et al. (2011).[6]