In mathematics, the Grothendieck inequality states that there is a universal constant with the following property. If Mij is an n × 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 constant being independent of n. For a fixed Hilbert space of dimension d, the smallest constant that satisfies this property for all n × n matrices is called a Grothendieck constant and denoted . In fact, there are two Grothendieck constants, and , depending on whether one works with real or complex numbers, respectively. The Grothendieck inequality and Grothendieck constants are named after Alexander Grothendieck, who proved the existence of the constants in a paper published in 1953. Motivation and the operator formulation Let be an matrix. Then defines a linear operator between the normed spaces and for . The -norm of is the quantity If , we denote the norm by . One can consider the following question: For what value of and is maximized? Since is linear, then it suffices to consider such that contains as many points as possible, and also such that is as large as possible. By comparing for , one sees that for all . One way to compute is by solving the following quadratic integer program: To see this, note that , and taking the maximum over gives . Then taking the maximum over gives by the convexity of and by the triangle inequality. This quadratic integer program can be relaxed to the following semidefinite program: It is known that exactly computing for is NP-hard, while exacting computing is NP-hard for . One can then ask the following natural question: How well does an optimal solution to the semidefinite program approximate ? The Grothendieck inequality provides an answer to this question: There exists a fixed constant such that, for any , for any matrix , and for any Hilbert space , Bounds on the constants The sequences and are easily seen to be increasing, and Grothendieck's result states that they are bounded, so they have limits. Grothendieck proved that where is defined to be . improved the result by proving that , conjecturing that the upper bound is tight. However, this conjecture was disproved by . Grothendieck constant of order d