{"id":5382,"date":"2025-03-04T13:57:35","date_gmt":"2025-03-04T05:57:35","guid":{"rendered":"https:\/\/transactions.nast.ph\/?p=5382"},"modified":"2025-04-23T13:32:32","modified_gmt":"2025-04-23T05:32:32","slug":"henstock-integration-in-a-hilbertian-countably-normed-space-with-nuclearity","status":"publish","type":"post","link":"https:\/\/transactions.nast.ph\/?p=5382","title":{"rendered":"Henstock Integration in a Hilbertian Countably Normed Space with Nuclearity"},"content":{"rendered":"\n

Sergio R. Canoy Jr.
Department of Mathematics, College of Science and Mathematics
MSU-Iligan Institute of Technology, Iligan City 9200<\/p>\n\n\n\n

http:\/\/doi.org\/10.57043\/transnastphl.2000.5382<\/a><\/p>\n\n\n\n

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Abstract<\/strong><\/p>\n\n\n\n

Henstock integration of real-valued functions has been extended to functions with values in normed spaces. Cao, who considered Banach-valued functions, showed that Henstock’s lemma\u2014which plays an important role in the real-valued case\u2014does not always hold in infinite-dimensional Banach spaces. Nakanishi showed that Henstock’s lemma holds in a ranked space called a Hilbertian CN-space with nuclearity.
In this paper, we revisit this space, define r-differentiability of a function with values in an r-separated ranked space, and present results concerning the primitives of Henstock integrable functions with values in this space. Furthermore, we provide a descriptive definition of the Henstock integral as defined by Nakanishi.<\/p>\n<\/div>\n\n\n\n

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Keywords<\/h5>\n\n\n

<\/span>AL-integral<\/a>, <\/span>Henstock integral<\/a>, <\/span>Hilbertian<\/a>, <\/span>nuclearity<\/a>, <\/span>r-differentiable<\/a>, <\/span>r-separated<\/a>, <\/span>ranked CN-space<\/a>, <\/span>strong Lusin<\/a><\/div><\/div>\n<\/div>\n\n\n\n
<\/object>TNP 2000 (22)_7_Henstock Integration in Hilbertian Space<\/a>Download<\/a><\/div>\n","protected":false},"excerpt":{"rendered":"

Sergio R. Canoy Jr.<\/p>\n","protected":false},"author":3,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[683,688],"tags":[705,702,704,706,708,707,703,709],"class_list":{"0":"post-5382","1":"post","2":"type-post","3":"status-publish","4":"format-standard","6":"category-683","7":"category-2000-technical-papers","8":"tag-al-integral","9":"tag-henstock-integral","10":"tag-hilbertian","11":"tag-nuclearity","12":"tag-r-differentiable","13":"tag-r-separated","14":"tag-ranked-cn-space","15":"tag-strong-lusin","16":"czr-hentry"},"_links":{"self":[{"href":"https:\/\/transactions.nast.ph\/index.php?rest_route=\/wp\/v2\/posts\/5382"}],"collection":[{"href":"https:\/\/transactions.nast.ph\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/transactions.nast.ph\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/transactions.nast.ph\/index.php?rest_route=\/wp\/v2\/users\/3"}],"replies":[{"embeddable":true,"href":"https:\/\/transactions.nast.ph\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=5382"}],"version-history":[{"count":3,"href":"https:\/\/transactions.nast.ph\/index.php?rest_route=\/wp\/v2\/posts\/5382\/revisions"}],"predecessor-version":[{"id":5463,"href":"https:\/\/transactions.nast.ph\/index.php?rest_route=\/wp\/v2\/posts\/5382\/revisions\/5463"}],"wp:attachment":[{"href":"https:\/\/transactions.nast.ph\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=5382"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/transactions.nast.ph\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=5382"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/transactions.nast.ph\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=5382"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}