why is grothendieck's inequality true?
August 17, 2026
The purpose of this post is to discuss Grothendieck’s inequality, which we state in the following form.
Theorem 1 (Grothendieck’s Theorem)
There is a universal constant \(K_G\) with the following property: let \(a_{ij}\) be an \(n\times n\) matrix and suppose that
\[\sup_{|s_i|\leq 1,\ |t_j|\leq 1} \left| \sum_{i,j=1}^n a_{ij}s_i t_j \right| \leq 1.\]Then
\[\sup_{\|x_i\|\leq 1,\ \|y_j\|\leq 1} \left| \sum_{i,j=1}^n a_{ij}\langle x_i,y_j\rangle_H \right| \leq K_G,\]where the \(x_i\) and \(y_j\) are vectors in a Hilbert space \(H\).