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Grothendieck inequality in nLab

  • ️Tue Apr 25 2017
Grothendieck inequality

Context

Analysis

Grothendieck inequality

Statement

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

Theorem

There exist sequences f n,g n:B→ℂf_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,y∈Bx, 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

General

Due to:

Review:

  • Leqi Zhu, Grothendieck’s inequality (2018) [pdf]

  • Ron Blei, Analysis in integer and fractional dimensions, Cambridge University Press (2009) [doi:10.1017/CBO9780511543012]

See also:

In quantum physics

Discussion of Grothendieck’s inequality in quantum physics, in relation to Bell's inequality, originates with:

reviewed in

Further discussion:

Last revised on December 23, 2022 at 10:44:50. See the history of this page for a list of all contributions to it.