{"paper":{"title":"Angles and Schauder basis in Hilbert spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Bingzhe Hou, Geng Tian, Xinzhi Zhang, Yang Cao","submitted_at":"2018-06-20T17:50:10Z","abstract_excerpt":"Let $\\mathcal{H}$ be a complex separable Hilbert space. We prove that if $\\{f_{n}\\}_{n=1}^{\\infty}$ is a Schauder basis of the Hilbert space $\\mathcal{H}$, then the angles between any two vectors in this basis must have a positive lower bound. Furthermore, we investigate that $\\{z^{\\sigma^{-1}(n)}\\}_{n=1}^{\\infty}$ can never be a Schauder basis of $L^{2}(\\mathbb{T},\\nu)$, where $\\mathbb{T}$ is the unit circle, $\\nu$ is a finite positive discrete measure, and $\\sigma: \\mathbb{Z} \\rightarrow \\mathbb{N}$ is an arbitrary surjective and injective map."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1806.07866","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}