\( \newcommand{\CC}{\mathbb{C}} \newcommand{\NN}{\mathbb{N}} \newcommand{\RR}{\mathbb{R}} \newcommand{\TT}{\mathbb{T}} \newcommand{\ZZ}{\mathbb{Z}} \newcommand{\H}{\mathcal{H}} \newcommand{\e}{\epsilon} \newcommand{\x}{\mathbf{x}} \newcommand{\y}{\mathbf{y}} \)

Grothendieck's Inequality

Statement

There exists a universal constant \(K\) with the following property. Whenever \((a_{i,j})\) is an \(n\times n\) complex matrix which satisfies \[ \left| \sum_{i,j} a_{i,j} x_i y_j \right| \le 1 \] for all sequences \(x_i\), \(y_j\) of complex numbers of modulus at most \(1\), then also \[ \left| \sum_{i,j} a_{i,j} \langle \x_i, \y_j \rangle \right| \le K \] for all sequences \(\x_i\), \(\y_j\) of vectors in the unit ball of a Hilbert space \(\H\).