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 \(H\).

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