Pre-Hilbert spaces without orthonormal bases

TL;DR

This paper characterizes pre-Hilbert spaces lacking orthonormal bases, demonstrating their existence for all uncountable dimensions and densities, and explores their properties through set-theoretic principles.

Contribution

It provides an algebraic characterization of such spaces and proves their existence across all uncountable cardinal pairs, including a Singular Compactness Theorem and a reflection principle.

Findings

Existence of pathological pre-Hilbert spaces for all uncountable dimensions and densities.

A set-theoretic characterization of when such spaces exist.

A reflection theorem linking subspaces to the Fodor-type Reflection Principle.

Abstract

We give an algebraic characterization of pre-Hilbert spaces with an orthonormal basis. This characterization is used to show that there are pre-Hilbert spaces $X$ of dimension and density $\lambda$ for any uncountable $\lambda$ without any orthonormal basis. Let us call a pre-Hilbert space without any orthonormal bases pathological. The pair of the cardinals $\kappa\leq\lambda$ such that there is a pre-Hilbert space of dimension $\kappa$ and density $\lambda$ are known to be characterized by the inequality $\lambda\leq\kappa^{\aleph_0}$. Our result implies that there are pathological pre-Hilbert spaces with dimension $\kappa$ and density $\lambda$ for all combinations of such $\kappa$ and $\lambda$ including the case $\kappa=\lambda$. A Singular Compactness Theorem on pathology of pre-Hilbert spaces is obtained. A reflection theorem asserting that for any pathological pre-Hilbert space $X$ there are stationarily many pathological sub-inner-product-spaces $Y$ of $X$ of smaller density is shown to be equivalent with Fodor-type Reflection Principle (FRP).