Math 2.0

Functional analysis Normed and Banach spaces Theorem

Grothendieck inequality

Builds on Hilbert spaceInner productProjective tensor product

Statement

There is a constant KK with the following property. Let m,n≥1m, n \ge 1 and let (aij)(a_{ij}) be a real m×nm \times n matrix. Assume that ∣∑i,jaijsitj∣≤1\abs{\sum_{i,j} a_{ij} s_i t_j} \le 1 for all s1,…,sm,t1,…,tn∈{−1,1}s_1, \dots, s_m, t_1, \dots, t_n \in \set{-1, 1}. Let HH be a real Hilbert space and let u1,…,um,v1,…,vnu_1, \dots, u_m, v_1, \dots, v_n be vectors of norm at most 11 in HH. Then ∣∑i=1m∑j=1naij⟨ui,vj⟩∣≤K.\Big| \sum_{i=1}^m \sum_{j=1}^n a_{ij} \ip{u_i}{v_j} \Big| \le K.

Remarks

The constant KK does not depend on mm, nn or HH. The smallest such constant is the real Grothendieck constant KGK_G. Its exact value is unknown. Krivine proved KG≤π/(2log⁡(1+2))≈1.78K_G \le \pi / (2 \log(1 + \sqrt{2})) \approx 1.78.

One proof uses random Gaussian vectors gg. For unit vectors uu and vv, the average of sgn⁡⟨g,u⟩sgn⁡⟨g,v⟩\sgn \ip{g}{u} \sgn \ip{g}{v} is 2πarcsin⁡⟨u,v⟩\frac{2}{\pi} \arcsin \ip{u}{v}. So signs drawn at random from the vectors reproduce a function of the inner products. Krivine first transforms the vectors so that this function becomes a fixed multiple of the inner product.

The inequality has a complex version, with a different constant. In computer science it shows that a semidefinite relaxation approximates the cut norm of a matrix within a constant factor.

Going deeper

For a student

Grothendieck stated the result in his work on tensor products of Banach spaces. In that language it says that every bounded linear map from L1L^1 to L2L^2 is absolutely summing. The inequality is a key tool in the study of the projective tensor product and of operator ideals.

Reading path

To understand this page from the ground up, read these 4 pages in order. The foundations come first.

Show the path as a graph
Hilbert space, concept, no evidence levelHilbert spaceconceptInner product, concept, no evidence levelInner productconceptProjective tensor product, concept, no evidence levelProjective tensor pro…conceptGrothendieck inequality, statement, no evidence levelGrothendieck inequali…statement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Hilbert space concept
  2. Inner product concept
  3. Projective tensor product concept
  4. Grothendieck inequality statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Grothendieck_inequality
About this page
TagNX8AMNXWG8
URIm2://NX8AMNXWG8
Kindstatement, theorem
WritingW0 stub
MSC46B28, 46M05, 46B07
StatusThis copy of the site has no build data. Run m2 status NX8AMNXWG8.
AboutHilbert space
UsesHilbert space, Inner product, Projective tensor product
Same as
Changed2 November 2026

Markdown · JSON · Edit on GitHub

What this page needs for W1

  • a source with a locator (0 of 1)

The writing levels say how complete the text of a page is. They do not measure its truth.

What links here

No page links to this page yet.