Grothendieck inequality

For a finite real scalar matrix $(a_{ij})$ and unit vectors $u_i,v_j$ in a real Hilbert space, $|\sum a_{ij}⟨u_i,v_j⟩|\le K_G{\max }_{{\epsilon }_i,{\delta }_j\in \{-1,1\}}|\sum a_{ij}{\epsilon }_i{\delta }_j|$, with a universal constant. One valid bound is $K_G\le \pi /(2\log (1+\sqrt{2}))$. Expand $\sin (ct)$ using odd tensor powers with coefficient magnitudes summing to $\sinh c=1$, then apply the Gaussian sign identity $\mathbb{E}[\mathrm{sgn}⟨G,u⟩\mathrm{sgn}⟨G,v⟩]=(2/\pi )\arcsin ⟨u,v⟩$. Taking $c=\log (1+\sqrt{2})$ 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.