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Continuity of Bounded Linear Operators on Normed Linear Spaces
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| Zeitschriftentitel: | Formalized Mathematics |
|---|---|
| Personen und Körperschaften: | , , |
| In: | Formalized Mathematics, 26, 2018, 3, S. 231-237 |
| Format: | E-Article |
| Sprache: | Englisch |
| veröffentlicht: |
Walter de Gruyter GmbH
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| Schlagwörter: |
| author_facet |
Nakasho, Kazuhisa Futa, Yuichi Shidama, Yasunari Nakasho, Kazuhisa Futa, Yuichi Shidama, Yasunari |
|---|---|
| author |
Nakasho, Kazuhisa Futa, Yuichi Shidama, Yasunari |
| spellingShingle |
Nakasho, Kazuhisa Futa, Yuichi Shidama, Yasunari Formalized Mathematics Continuity of Bounded Linear Operators on Normed Linear Spaces Applied Mathematics Computational Mathematics |
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nakasho, kazuhisa |
| spelling |
Nakasho, Kazuhisa Futa, Yuichi Shidama, Yasunari 1898-9934 1426-2630 Walter de Gruyter GmbH Applied Mathematics Computational Mathematics http://dx.doi.org/10.2478/forma-2018-0021 <jats:title>Summary</jats:title> <jats:p>In this article, using the Mizar system [1], [2], we discuss the continuity of bounded linear operators on normed linear spaces. In the first section, it is discussed that bounded linear operators on normed linear spaces are uniformly continuous and Lipschitz continuous. Especially, a bounded linear operator on the dense subset of a complete normed linear space has a unique natural extension over the whole space. In the next section, several basic currying properties are formalized.</jats:p> <jats:p>In the last section, we formalized that continuity of bilinear operator is equivalent to both Lipschitz continuity and local continuity. We referred to [4], [13], and [3] in this formalization.</jats:p> Continuity of Bounded Linear Operators on Normed Linear Spaces Formalized Mathematics |
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Walter de Gruyter GmbH, 2018 |
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Walter de Gruyter GmbH, 2018 |
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1898-9934 1426-2630 |
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Walter de Gruyter GmbH |
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Formalized Mathematics |
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49 |
| title |
Continuity of Bounded Linear Operators on Normed Linear Spaces |
| title_unstemmed |
Continuity of Bounded Linear Operators on Normed Linear Spaces |
| title_full |
Continuity of Bounded Linear Operators on Normed Linear Spaces |
| title_fullStr |
Continuity of Bounded Linear Operators on Normed Linear Spaces |
| title_full_unstemmed |
Continuity of Bounded Linear Operators on Normed Linear Spaces |
| title_short |
Continuity of Bounded Linear Operators on Normed Linear Spaces |
| title_sort |
continuity of bounded linear operators on normed linear spaces |
| topic |
Applied Mathematics Computational Mathematics |
| url |
http://dx.doi.org/10.2478/forma-2018-0021 |
| publishDate |
2018 |
| physical |
231-237 |
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<jats:title>Summary</jats:title>
<jats:p>In this article, using the Mizar system [1], [2], we discuss the continuity of bounded linear operators on normed linear spaces. In the first section, it is discussed that bounded linear operators on normed linear spaces are uniformly continuous and Lipschitz continuous. Especially, a bounded linear operator on the dense subset of a complete normed linear space has a unique natural extension over the whole space. In the next section, several basic currying properties are formalized.</jats:p>
<jats:p>In the last section, we formalized that continuity of bilinear operator is equivalent to both Lipschitz continuity and local continuity. We referred to [4], [13], and [3] in this formalization.</jats:p> |
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| author | Nakasho, Kazuhisa, Futa, Yuichi, Shidama, Yasunari |
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| description | <jats:title>Summary</jats:title> <jats:p>In this article, using the Mizar system [1], [2], we discuss the continuity of bounded linear operators on normed linear spaces. In the first section, it is discussed that bounded linear operators on normed linear spaces are uniformly continuous and Lipschitz continuous. Especially, a bounded linear operator on the dense subset of a complete normed linear space has a unique natural extension over the whole space. In the next section, several basic currying properties are formalized.</jats:p> <jats:p>In the last section, we formalized that continuity of bilinear operator is equivalent to both Lipschitz continuity and local continuity. We referred to [4], [13], and [3] in this formalization.</jats:p> |
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| imprint | Walter de Gruyter GmbH, 2018 |
| imprint_str_mv | Walter de Gruyter GmbH, 2018 |
| institution | DE-Gla1, DE-Zi4, DE-15, DE-Pl11, DE-Rs1, DE-105, DE-14, DE-Ch1, DE-L229, DE-D275, DE-Bn3, DE-Brt1, DE-Zwi2, DE-D161 |
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| physical | 231-237 |
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| series | Formalized Mathematics |
| source_id | 49 |
| spelling | Nakasho, Kazuhisa Futa, Yuichi Shidama, Yasunari 1898-9934 1426-2630 Walter de Gruyter GmbH Applied Mathematics Computational Mathematics http://dx.doi.org/10.2478/forma-2018-0021 <jats:title>Summary</jats:title> <jats:p>In this article, using the Mizar system [1], [2], we discuss the continuity of bounded linear operators on normed linear spaces. In the first section, it is discussed that bounded linear operators on normed linear spaces are uniformly continuous and Lipschitz continuous. Especially, a bounded linear operator on the dense subset of a complete normed linear space has a unique natural extension over the whole space. In the next section, several basic currying properties are formalized.</jats:p> <jats:p>In the last section, we formalized that continuity of bilinear operator is equivalent to both Lipschitz continuity and local continuity. We referred to [4], [13], and [3] in this formalization.</jats:p> Continuity of Bounded Linear Operators on Normed Linear Spaces Formalized Mathematics |
| spellingShingle | Nakasho, Kazuhisa, Futa, Yuichi, Shidama, Yasunari, Formalized Mathematics, Continuity of Bounded Linear Operators on Normed Linear Spaces, Applied Mathematics, Computational Mathematics |
| title | Continuity of Bounded Linear Operators on Normed Linear Spaces |
| title_full | Continuity of Bounded Linear Operators on Normed Linear Spaces |
| title_fullStr | Continuity of Bounded Linear Operators on Normed Linear Spaces |
| title_full_unstemmed | Continuity of Bounded Linear Operators on Normed Linear Spaces |
| title_short | Continuity of Bounded Linear Operators on Normed Linear Spaces |
| title_sort | continuity of bounded linear operators on normed linear spaces |
| title_unstemmed | Continuity of Bounded Linear Operators on Normed Linear Spaces |
| topic | Applied Mathematics, Computational Mathematics |
| url | http://dx.doi.org/10.2478/forma-2018-0021 |