Statement of The Theorem
Let (V, b) be a finite-dimensional vector space over an arbitrary field k together with a nondegenerate symmetric or skew-symmetric bilinear form. If f: U→U' is an isometry between two subspaces of V then f extends to an isometry of V.
Witt's theorem implies that the dimension of a maximal isotropic subspace of V is an invariant, called the index or Witt index of b, and moreover, that the isometry group of (V, b) acts transitively on the set of maximal isotropic subspaces. This fact plays an important role in the structure theory and representation theory of the isometry group and in the theory of reductive dual pairs.
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