Grothendieck inequality

In mathematics, the Grothendieck inequality states that there is a universal constant ${\displaystyle K_{G}}$ with the following property. If M_ij is an n × n (real or complex) matrix with

${\displaystyle {\Big |}\sum _{i,j}M_{ij}s_{i}t_{j}{\Big |}\leq 1}$

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

${\displaystyle {\Big |}\sum _{i,j}M_{ij}\langle S_{i},T_{j}\rangle {\Big |}\leq K_{G}}$

for all vectors S_i, T_j in the unit ball B(H) of a (real or complex) Hilbert space H, the constant ${\displaystyle K_{G}}$ 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 ${\displaystyle K_{G}(d)}$. In fact, there are two Grothendieck constants, ${\displaystyle K_{G}^{\mathbb {R} }(d)}$ and ${\displaystyle K_{G}^{\mathbb {C} }(d)}$, 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.