Dvoretzky's theorem

Every infinite-dimensional normed space contains finite-dimensional subspaces of arbitrarily large dimension whose inherited norms are arbitrarily close to Euclidean.

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Banach space: A complete normed vector space. Dvoretzky's theorem applies to infinite-dimensional normed spaces, including Banach spaces.
Banach–Mazur distance: A multiplicative measure of how far apart two finite-dimensional normed spaces are up to linear isomorphism. It expresses precisely how close a subspace's norm is to the Euclidean norm.
A. Dvoretzky: A mathematician whose work established a fundamental theorem on Euclidean subspaces of normed spaces. His name identifies the theorem and its original finite-dimensional formulation.
Dvoretzky–Rogers theorem: A theorem stating that every infinite-dimensional Banach space has an unconditionally convergent series that is not absolutely convergent. Despite the similar name, it concerns series convergence rather than Euclidean subspaces.
Euclidean space: A real vector space equipped with the norm induced by an inner product. The theorem identifies large subspaces whose geometry approximates this familiar model.
Euclidean section: A subspace of a normed space whose inherited norm is Euclidean or approximately Euclidean. Such sections are the geometric objects whose dimensions Dvoretzky's theorem guarantees.
Dvoretsky–Rogers lemma: A result in Banach space theory concerning summability and weakly unconditionally Cauchy series. It is another foundational theorem associated with Dvoretzky's contributions to Banach space geometry.
Kashin's theorem: A result showing that some subspaces of finite-dimensional ℓ₁ have norms uniformly equivalent to Euclidean norms. It gives a particularly strong finite-dimensional construction related to Dvoretzky's phenomenon.
Isometric embedding: A map between metric spaces that preserves all distances. Exact isometric copies of Euclidean spaces are stronger than the theorem's approximate copies.
Dvoretzky dimension: The largest dimension of a nearly Euclidean subspace obtainable at a specified distortion. It quantifies the dimension-versus-accuracy tradeoff in finite-dimensional versions of the theorem.