Math 2.0

Functional analysis Normed and Banach spaces Theorem

Dvoretzky's theorem

Builds on Normed vector spaceLinear subspaceNorm induced by an inner productType and cotypeBanach–Mazur compactum

Statement

Let k≥1k \ge 1 be an integer and let ε>0\varepsilon > 0. Then there is an integer N=N(k,ε)N = N(k, \varepsilon) with the following property. Let XX be a real normed space of dimension at least NN. Then XX has a kk-dimensional linear subspace EE with a linear map T ⁣:ℝk→ET \colon \R^k \to E such that ∣x∣≤∥Tx∥≤(1+ε)∣x∣for all x∈ℝk,\abs{x} \le \norm{T x} \le (1 + \varepsilon) \abs{x} \quad \text{for all } x \in \R^k, where ∣x∣\abs{x} is the Euclidean norm of xx.

Remarks

The theorem was proved by Dvoretzky. V. Milman gave a proof based on concentration of measure on the high-dimensional sphere. His proof shows that one can take kk of order c(ε)log⁡nc(\varepsilon) \log n in a space of dimension nn. The idea: a norm, seen as a Lipschitz function on the Euclidean sphere, is almost constant on most of the sphere. A random kk-dimensional subspace then sees only this almost constant part.

The bound log⁡n\log n cannot be improved in general. For ℓ∞n\ell_\infty^n, that is ℝn\R^n with the max norm, the largest kk with a subspace at distance at most 22 from Euclidean is of order log⁡n\log n. For ℓ1n\ell_1^n, in contrast, there are almost Euclidean subspaces of dimension proportional to nn.

Going deeper

For a student

In the language of the local theory of Banach spaces, the theorem says that ℓ2\ell_2 is finitely representable in every infinite-dimensional Banach space. It is a starting point of asymptotic geometric analysis, together with the notions of type and cotype. The distance used here is the Banach–Mazur distance of the Banach–Mazur compactum.

Reading path

To understand this page from the ground up, read these 7 pages in order. The foundations come first.

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Linear subspace, concept, no evidence levelLinear subspaceconceptNorm induced by an inner product, concept, no evidence levelNorm induced by an in…conceptNormed vector space, concept, no evidence levelNormed vector spaceconceptOperator norm, concept, no evidence levelOperator normconceptType and cotype, concept, no evidence levelType and cotypeconceptBanach–Mazur compactum, object, no evidence levelBanach–Mazur compactumobjectDvoretzky's theorem, statement, no evidence levelDvoretzky's theoremstatement
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. Linear subspace concept
  2. Norm induced by an inner product concept
  3. Normed vector space concept
  4. Operator norm concept
  5. Type and cotype concept
  6. Banach–Mazur compactum object
  7. Dvoretzky's theorem statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Dvoretzky%27s_theorem
About this page
TagNRPQP49N9F
URIm2://NRPQP49N9F
Kindstatement, theorem
WritingW0 stub
MSC46B07, 52A21, 46B09
StatusThis copy of the site has no build data. Run m2 status NRPQP49N9F.
AboutNormed vector space
UsesNormed vector space, Linear subspace, Norm induced by an inner product, Type and cotype, Banach–Mazur compactum
Same as
Changed2 November 2026

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