Proszę używać tego identyfikatora do cytowań lub wstaw link do tej pozycji:
http://hdl.handle.net/11320/4898
Tytuł: | The Orthogonal Projection and the Riesz Representation Theorem |
Autorzy: | Narita, Keiko Endou, Noboru Shidama, Yasunari |
Słowa kluczowe: | normed linear spaces Banach spaces duality orthogonal projection Riesz representation |
Data wydania: | 2015 |
Data dodania: | 16-gru-2016 |
Wydawca: | De Gruyter Open |
Źródło: | Formalized Mathematics, Volume 23, Issue 3, 243–252 |
Abstrakt: | AbstractIn this article, the orthogonal projection and the Riesz representation theorem are mainly formalized. In the first section, we defined the norm of elements on real Hilbert spaces, and defined Mizar functor RUSp2RNSp, real normed spaces as real Hilbert spaces. By this definition, we regarded sequences of real Hilbert spaces as sequences of real normed spaces, and proved some properties of real Hilbert spaces. Furthermore, we defined the continuity and the Lipschitz the continuity of functionals on real Hilbert spaces. Referring to the article [15], we also defined some definitions on real Hilbert spaces and proved some theorems for defining dual spaces of real Hilbert spaces. As to the properties of all definitions, we proved that they are equivalent properties of functionals on real normed spaces. In Sec. 2, by the definitions [11], we showed properties of the orthogonal complement. Then we proved theorems on the orthogonal decomposition of elements of real Hilbert spaces. They are the last two theorems of existence and uniqueness. In the third and final section, we defined the kernel of linear functionals on real Hilbert spaces. By the last three theorems, we showed the Riesz representation theorem, existence, uniqueness, and the property of the norm of bounded linear functionals on real Hilbert spaces. We referred to [36], [9], [24] and [3] in the formalization. |
Afiliacja: | Keiko Narita - Hirosaki-city, Aomori, Japan Noboru Endou - Gifu National College of Technology, Gifu, Japan Yasunari Shidama - Shinshu University, Nagano, Japan |
URI: | http://hdl.handle.net/11320/4898 |
DOI: | 10.1515/forma-2015-0020 |
ISSN: | 1426-2630 1898-9934 |
Typ Dokumentu: | Article |
Występuje w kolekcji(ach): | Formalized Mathematics, 2015, Volume 23, Issue 3 |
Pliki w tej pozycji:
Plik | Opis | Rozmiar | Format | |
---|---|---|---|---|
forma-2015-0020.pdf | 262,07 kB | Adobe PDF | Otwórz |
Pozycja ta dostępna jest na podstawie licencji Licencja Creative Commons CCL