2016/06/13 by Sakaé Fuchino, Fuchino, Sakaé · 2 citations
Mathematics · #03E75 #46C05 #Advanced Banach Space Theory #Advanced Topics in Algebra #Algebraic number #Basis (linear algebra) #Characterization (materials science) #Countable set #Dimension (graph theory) #FOS: Mathematics #Geometry #Hilbert manifold #Hilbert space #Inner product space #Logic (math.LO) #Mathematical Analysis and Transform Methods #Mathematical analysis #Mathematics #Orthonormal basis #Physics #Pure mathematics #Quantum mechanics #Reflection principle (Wiener process) #Reproducing kernel Hilbert space #Rigged Hilbert space #Space (punctuation) #Uncountable set #math.LO #msc:03E75 #msc:46C05
paper · pdf · doi:10.48550/arxiv.1606.03869
published in arXiv (Cornell University) (Cornell University)
openalex publication_date 2016/06/13 · arxiv created 2016/06/27 · arxiv updated 2016/06/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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≤λ such that there is a pre-Hilbert space of dimension κ and density λ are known to be characterized by the inequality λ≤κℵ0. 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).