Affine Geometries of Paths Possessing an Invariant Integral by Eisenhart L. P.

By Eisenhart L. P.

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3 is not first-separable. Notice that in first-separable spaces the involved neighbourhoods are not necessarily disjoint. If we require the existence of disjoint neighbourhoods for every two points, we have second-separability, a property more commonly named after Hausdorff. 15 Hausdorff character A topological space S is said to be a Hausdorff space if every two distinct points p, q ∈ S have disjoint neighbourhoods. There are consequently U p and V q such that U ∩ V = ∅. 3). 18). Another non-Hausdorff space is given by two copies of 24 CHAPTER 1.

9), which makes integration on the group possible. Such measures, which are unique up to real positive factors, are essential to the theory of group representations and general Fourier analysis. Unlike finite-dimensional euclidean spaces, Hilbert spaces are not locally compact. They are infinite-dimensional, and there are fundamental differences between finite-dimensional and infinite-dimensional spaces. One of the main distinctive properties comes out precisely here: Riesz theorem: a normed vector space is locally compact if and only if its dimension is finite.

Other topologies may be defined on function spaces and their choice is a matter of convenience. A point worth emphasizing is that, besides other requirements, a topology is presupposed in functional differentiation and integration. 13, there is one remarkable difference: when the topology is given by a norm, an infinite-dimensional space is never locally compact. 15 Connection by paths A topological space S is path-connected (or arcwise-connected ) if, for every two points p, q in S there exists a path α with α(0) = p and α(1) = q.

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