Achievements on Matrix Hilbert spaces and reproducing kernel matrix Hilbert spaces

Achievements on Matrix Hilbert spaces and reproducing kernel matrix Hilbert spaces | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article Achievements on Matrix Hilbert spaces and reproducing kernel matrix Hilbert spaces Elnaz Osgooei This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-3326481/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract ‎Hilbert space is a very powerfull mathematical tool that has proven to be incredibily useful in a wide range of applications‎. ‎Matrix Hilebrt space is a new frame work that has been presented recently to perform a matrix inner product space when data observation is proposed as matrices and it is needed to analyse large dataset with complex and multi-dimensional structure‎. ‎In this paper‎, ‎a new norm associated to sequences in matrix Hilbert spaces is defined and the relation between this new norm and the norm obtained by matrix inner product is investigated‎. ‎Also‎,‎it is discussed that although‎, ‎some results and concepts in Hilbert spaces can be extended to the matrix Hilbert spaces‎, ‎but counterexamples show that the results like Pythagorean theorem‎, ‎Parallelogram law‎, ‎Jordan-Von Neumann theorem are not hold in matrix Hilbert spaces‎. ‎At last the reproducing kernel Hilbert space (RKHS) is extended to reproducing kernel matrix Hilbert space (RKMHS) and by considering some appropriate conditions on a matrix Hilbert space‎, ‎an explicit structure of reproducing kernel is obtained and the RKMHS‎, ‎H(K)‎, ‎which is obtained by reproducing kernel $K=K_{1}+K_{2}$ is characterized‎. Matrix Hilbert spaces Matrix inner product spaces Reproducing kernel matrix Hilbert spaces Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Posted Version 1 posted You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. Our growing team is made up of researchers and industry professionals working together to solve the most critical problems facing scientific publishing. Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-3326481","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":236536346,"identity":"98ddaf9c-9815-47c0-b416-d3aba69086b5","order_by":0,"name":"Elnaz 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