The topological spaces that support Haar systems

Abstract

ABSTRACT. A topological space X supports a nontrivial Haar system if and only if there is a continuous one-to-one map of X into the unit circle. This gives an elementary proof of Mairhuber's classical theorem for compact X. It answers a 1960 question asked by R. R. Phelps when X is locally compact.

On a hierarchy of infinite-dimensional spaces and related Kolmogorov-Gelfand widths

“…Let us mention that the space X N generated by a Chebyshev system is often called Haar space, cf. [27]. Thus one may also say that a generic N−dimensional subspace of C N −1 (I) is piecewise Haar space.…”

“…By a technique similar to the already used we may prove the following results which generalize Theorem 22. We assume that K * p is the set defined by (27) with a strongly elliptic constant coefficients operator L 2p . The space X p is defined by (40) and the space F L by (41).…”

Multidimensional Chebyshev spaces, hierarchy of infinite-dimensional spaces and Kolmogorov-Gelfand widths

“…They were further developed and applied to the generalized Moment problem and Approximation theory by A. Haar (the Haar spaces), S. Bernstein, M. Krein, S. Karlin, and others, cf. [29], [20], [34]. 2 In general, zero set properties and intersections are not a reliable reference point for multidimensional Analysis.…”

“…Assume that the differential operators P 2M and P ′ 2N , associated with X M and X N , have trivial factorization operators by the definition (35), and trivial domain partitions D = D 1 and D = D ′ 1 by (34). Then the space of solutions of the Elliptic BVP (44)-( 46) where G = D is a subspace of the space…”