## Zentralblatt MATH

Publications of (and about) Paul Erdös

Zbl.No:  025.18701
Autor:  Erdös, Paul
Title:  The dimension of the rational points in Hilbert space. (In English)
Source:  Ann. of Math., II. Ser. 41, 734-736 (1940).
Review:  Let H denote the Hilbert space of all sequences of real numbers (x1,x2,...) such that sumi = 1oo x2i < oo. Let R be the set of points of H having all coordinates rational. Let R0 be the set of points of H of the form ({1\over n1}, {1\over n2}, ··· ), where the ni are positive integers. Let R1 be the closure of R0. The author shows that R0,R1 and R have the dimension 1.
As the cartesian product R1 × R1 is homeomorphic to R1, it follows that there exists a metric separable complete space X such that X and X × X have dimension 1.
Reviewer:  Béla de.Sz.Nagy (Szeged)
Classif.:  * 46C99 Inner product spaces, Hilbert spaces
Index Words:  Functional analysis

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