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**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 (x_{1},x_{2},...) such that **sum**_{i = 1}^{oo} x^{2}_{i} < oo. Let R be the set of points of H having all coordinates rational. Let R_{0} be the set of points of H of the form **(**{1\over n_{1}}, {1\over n_{2}}, ··· **)**, where the n_{i} are positive integers. Let R_{1} be the closure of R_{0}. The author shows that R_{0},R_{1} and R have the dimension 1.

As the cartesian product R_{1} × R_{1} is homeomorphic to R_{1}, 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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