= Grothendieck inequality
{c}
For a finite real scalar matrix $(a_{ij})$ and unit vectors $u_i,v_j$ in a real <Hilbert space>, $|\sum a_{ij}\langle u_i,v_j\rangle|\le K_G\max_{\varepsilon_i,\delta_j\in\{-1,1\}}|\sum a_{ij}\varepsilon_i\delta_j|$, with a universal constant. One valid bound is $K_G\le\pi/(2\log(1+\sqrt2))$. Expand $\sin(ct)$ using odd <tensor powers> with coefficient magnitudes summing to $\sinh c=1$, then apply the Gaussian sign identity $\mathbb E[\operatorname{sgn}\langle G,u\rangle\operatorname{sgn}\langle G,v\rangle]=(2/\pi)\arcsin\langle u,v\rangle$. Taking $c=\log(1+\sqrt2)$ converts the expectation back to the original inner product. Complex scalars admit a universal bound twice this real bound by separating real and imaginary parts.
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