nLab
Grothendieck inequality

Grothendieck inequality

Statement

Let BB be the unit ball of a separable Hilbert space over the real or numbers. Then the scalar product, ,:B×B\langle\cdot,\cdot\rangle : B \times B \to \mathbb{C} has the following special property:

Theorem

There exist sequences f n,g n:Bf_n,g_n: B \to \mathbb{C} of norm-continuous functions, such that * x,y= nf n(x)g n(y)\langle x, y \rangle = \sum_n f_n(x) g_n(y) for all x,yBx, y \in B * nsup B|f n|sup B|g n|<\sum_n \sup_B \left| f_n \right| \sup_B \left| g_n \right| &lt; \infty

In other words, ,\langle\cdot,\cdot\rangle, as a function of two variables, is an element of the projective tensor product C(B)^C(B)C(B) {\displaystyle\hat{\otimes}} C(B). Its projective tensor norm is known as Grothendieck’s constant. The precise value of this constant is different in the real and complex case, and neither one is known exactly.

References

A very nice proof and some applications of this result are discussed in