Pre-Hilbert spaces without orthonormal bases

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 $λ$ for any uncountable $λ$ without any orthonormal basis. Let us call a pre-Hilbert space without any orthonormal bases pathological. The pair of the cardinals $κ\leqλ$ such that there is a pre-Hilbert space of dimension $κ$ and density $λ$ are known to be characterized by the inequality $λ\leqκ^{\aleph_0}$. Our result implies that there are pathological pre-Hilbert spaces with dimension $κ$ and density $λ$ for all combinations of such $κ$ and $λ$ including the case $κ=λ$. 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).