{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,4,2]],"date-time":"2022-04-02T21:38:42Z","timestamp":1648935522980},"reference-count":32,"publisher":"Walter de Gruyter GmbH","issue":"3","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2010,1,1]]},"abstract":"<jats:title>On <jats:italic>L<jats:sup>p<\/jats:sup><\/jats:italic> Space Formed by Real-Valued Partial Functions<\/jats:title>\n        <jats:p>This article is the continuation of [31]. We define the set of <jats:italic>L<jats:sup>p<\/jats:sup><\/jats:italic> integrable functions - the set of all partial functions whose absolute value raised to the <jats:italic>p<\/jats:italic>-th power is integrable. We show that <jats:italic>L<jats:sup>p<\/jats:sup><\/jats:italic> integrable functions form the <jats:italic>L<jats:sup>p<\/jats:sup><\/jats:italic> space. We also prove Minkowski's inequality, H\u00f6lder's inequality and that <jats:italic>L<jats:sup>p<\/jats:sup><\/jats:italic> space is Banach space ([15], [27]).<\/jats:p>","DOI":"10.2478\/v10037-010-0018-6","type":"journal-article","created":{"date-parts":[[2011,1,6]],"date-time":"2011-01-06T02:22:44Z","timestamp":1294280564000},"page":"159-169","source":"Crossref","is-referenced-by-count":0,"title":["On Lp Space Formed by Real-Valued Partial Functions"],"prefix":"10.2478","volume":"18","author":[{"given":"Yasushige","family":"Watase","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Noboru","family":"Endou","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yasunari","family":"Shidama","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"374","reference":[{"issue":"3","key":"1","first-page":"565","article-title":"Ring ideals","volume":"9","author":"Jonathan Backer","year":"2001","journal-title":"Formalized Mathematics"},{"issue":"1","key":"2","first-page":"41","article-title":"The fundamental properties of natural numbers","volume":"1","author":"Grzegorz Bancerek","year":"1990","journal-title":"Formalized Mathematics"},{"issue":"1","key":"3","first-page":"91","article-title":"The ordinal numbers","volume":"1","author":"Grzegorz Bancerek","year":"1990","journal-title":"Formalized Mathematics"},{"issue":"4","key":"4","first-page":"485","article-title":"Miscellaneous facts about functions","volume":"5","author":"Grzegorz Bancerek","year":"1996","journal-title":"Formalized Mathematics"},{"issue":"1","key":"5","first-page":"173","article-title":"Series of positive real numbers. 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