Instead, with the topology of compact convergence, C can be given the structure of a Fr�chet space : a locally convex topological vector space whose topology can be induced by a complete translation-invariant metric.
12.
Indeed, a topological vector space is called complete iff its uniformity ( induced by its topology and addition operation ) is complete; the uniformity induced by a translation-invariant metric that induces the topology coincides with the original uniformity.
13.
Born and H . S . Green similarly introduced the notion an invariant ( quantum ) metric operator x _ k x ^ k + p _ k p ^ k as extension of the Minkowski metric of special relativity to an invariant metric on phase space coordinates.
14.
*The space of measurable functions on the unit interval ( where we identify two functions that are equal almost everywhere ) has a vector-space topology defined by the translation-invariant metric : ( which induces the convergence in measure of measurable functions; for random variables, convergence in measure is convergence in probability)
15.
More generally, whenever one has a compact group with Haar measure ?, and an arbitrary inner product " h ( X, Y ) " defined at the tangent space of some point in " G ", one can define an invariant metric simply by averaging over the entire group, i . e . by defining
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If G is not compact it is not definite and hence not an inner product : however when G is semisimple and K is a maximal compact subgroup it can be used to define a G-invariant metric on the homogeneous space X = G / K : such Riemannian manifolds are called symmetric spaces of non-compact type without Euclidean factors.
17.
The developments in Computational Anatomy included the establishment of the Sobelev smoothness conditions on the diffeomorphometry metric to insure existence of solutions of variational problems in the space of diffeomorphisms, the derivation of the Euler-Lagrange equations characterizing geodesics through the group and associated conservation laws, the demonstration of the metric properties of the right invariant metric, the demonstration that the Euler-Lagrange equations have a well-posed initial value problem with unique solutions for all time, and with the first results on sectional curvatures for the diffeomorphometry metric in landmarked spaces.
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