There exist sequences$f_n,g_n: B \to \mathbb{C}$ of norm-continuous functions, such that * $\langle x, y \rangle = \sum_n f_n(x) g_n(y)$ for all $x, y \in B$ * $\sum_n \sup_B \left| f_n \right| \sup_B \left| g_n \right| < \infty$

In other words, $\langle\cdot,\cdot\rangle$, as a function of two variables, is an element of the projective tensor product$C(B) {\displaystyle\hat{\otimes}} C(B)$. 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

Ron Blei, Analysis in integer and fractional dimensions.