Lipschitz tensor product
Inspired by ideas of R. Schatten in his celebrated monograph on a theory of cross-spaces, we introduce the notion of a Lipschitz tensor product X\boxtimes E of a pointed metric space and a Banach space E as a certain linear subspace of the algebraic dual of Lipo(X,E^*). We prove that forms a dual p...
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| creator | Cabrera-Padilla, M. G Chávez-Domínguez, J. A Jiménez-Vargas, A Villegas-Vallecillos, Moisés |
| description | Inspired by ideas of R. Schatten in his celebrated monograph on a theory of
cross-spaces, we introduce the notion of a Lipschitz tensor product X\boxtimes
E of a pointed metric space and a Banach space E as a certain linear subspace
of the algebraic dual of Lipo(X,E^*). We prove that
forms a dual pair. We prove that X\boxtimes E is linearly isomorphic to the
linear space of all finite-rank continuous linear operators from (X^#,T) into
E, where X^# denotes the space Lipo(X,K) and T is the topology of pointwise
convergence of X^#. The concept of Lipschitz tensor product of elements of X^#
and E^* yields the space X^#\boxast E^* as a certain linear subspace of the
algebraic dual of X\boxtimes E. To ensure the good behavior of a norm on
X\boxtimes E with respect to the Lipschitz tensor product of Lipschitz
functionals (mappings) and bounded linear functionals (operators), the concept
of dualizable (respectively, uniform) Lipschitz cross-norm on X\boxtimes E is
defined. We show that the Lipschitz injective norm epsilon, the Lipschitz
projective norm pi and the Lipschitz p-nuclear norm d_p (1 |
| doi_str_mv | 10.48550/arxiv.1408.1874 |
| format | Article |
| fullrecord | <record><control><sourceid>arxiv_GOX</sourceid><recordid>TN_cdi_arxiv_primary_1408_1874</recordid><sourceformat>XML</sourceformat><sourcesystem>PC</sourcesystem><sourcerecordid>1408_1874</sourcerecordid><originalsourceid>FETCH-LOGICAL-a654-f4f055ac94abf6ad51cab96ad394493964e1a6fd0668313995ccad7144f0eed33</originalsourceid><addsrcrecordid>eNotzjsPgjAUBeAuDgbdmQx_AGzT20JHQ3wlJC7s5NJHJFEhBY366wV1Omc5Jx8hIaMJZELQNfpn80gY0CxhWQpzEhZN1-tzM7yjwd761kedb81dDwsyc3jp7fKfASl32zI_xMVpf8w3RYxSQOzAUSFQK8DaSTSCaazVWLgCUFxJsAylM1TKjDOulNAaTcpg3FlrOA_I6nf7lVWdb67oX9UkrCYh_wB7ZTU-</addsrcrecordid><sourcetype>Open Access Repository</sourcetype><iscdi>true</iscdi><recordtype>article</recordtype></control><display><type>article</type><title>Lipschitz tensor product</title><source>arXiv.org</source><creator>Cabrera-Padilla, M. G ; Chávez-Domínguez, J. A ; Jiménez-Vargas, A ; Villegas-Vallecillos, Moisés</creator><creatorcontrib>Cabrera-Padilla, M. G ; Chávez-Domínguez, J. A ; Jiménez-Vargas, A ; Villegas-Vallecillos, Moisés</creatorcontrib><description>Inspired by ideas of R. Schatten in his celebrated monograph on a theory of
cross-spaces, we introduce the notion of a Lipschitz tensor product X\boxtimes
E of a pointed metric space and a Banach space E as a certain linear subspace
of the algebraic dual of Lipo(X,E^*). We prove that <Lipo(X,E^*),X\boxtimes E>
forms a dual pair. We prove that X\boxtimes E is linearly isomorphic to the
linear space of all finite-rank continuous linear operators from (X^#,T) into
E, where X^# denotes the space Lipo(X,K) and T is the topology of pointwise
convergence of X^#. The concept of Lipschitz tensor product of elements of X^#
and E^* yields the space X^#\boxast E^* as a certain linear subspace of the
algebraic dual of X\boxtimes E. To ensure the good behavior of a norm on
X\boxtimes E with respect to the Lipschitz tensor product of Lipschitz
functionals (mappings) and bounded linear functionals (operators), the concept
of dualizable (respectively, uniform) Lipschitz cross-norm on X\boxtimes E is
defined. We show that the Lipschitz injective norm epsilon, the Lipschitz
projective norm pi and the Lipschitz p-nuclear norm d_p (1<=p<=infty) are
uniform dualizable Lipschitz cross-norms on X\boxtimes E. In fact, epsilon is
the least dualizable Lipschitz cross-norm and pi is the greatest Lipschitz
cross-norm on X\boxtimes E. Moreover, dualizable Lipschitz cross-norms alpha on
X\boxtimes E are characterized by satisfying the relation epsilon<=alpha<=pi.
In addition, the Lipschitz injective (projective) norm on X\boxtimes E can be
identified with the injective (respectively, projective) tensor norm on the
Banach-space tensor product between the Lipschitz-free space over X and E. In
terms of the space X^#\boxast E^*, we describe the spaces of Lipschitz compact
(finite-rank, approximable) operators from X to E^$.</description><identifier>DOI: 10.48550/arxiv.1408.1874</identifier><language>eng</language><subject>Mathematics - Functional Analysis</subject><creationdate>2014-08</creationdate><rights>http://arxiv.org/licenses/nonexclusive-distrib/1.0</rights><oa>free_for_read</oa><woscitedreferencessubscribed>false</woscitedreferencessubscribed></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><link.rule.ids>228,230,780,885</link.rule.ids><linktorsrc>$$Uhttps://arxiv.org/abs/1408.1874$$EView_record_in_Cornell_University$$FView_record_in_$$GCornell_University$$Hfree_for_read</linktorsrc><backlink>$$Uhttps://doi.org/10.48550/arXiv.1408.1874$$DView paper in arXiv$$Hfree_for_read</backlink></links><search><creatorcontrib>Cabrera-Padilla, M. G</creatorcontrib><creatorcontrib>Chávez-Domínguez, J. A</creatorcontrib><creatorcontrib>Jiménez-Vargas, A</creatorcontrib><creatorcontrib>Villegas-Vallecillos, Moisés</creatorcontrib><title>Lipschitz tensor product</title><description>Inspired by ideas of R. Schatten in his celebrated monograph on a theory of
cross-spaces, we introduce the notion of a Lipschitz tensor product X\boxtimes
E of a pointed metric space and a Banach space E as a certain linear subspace
of the algebraic dual of Lipo(X,E^*). We prove that <Lipo(X,E^*),X\boxtimes E>
forms a dual pair. We prove that X\boxtimes E is linearly isomorphic to the
linear space of all finite-rank continuous linear operators from (X^#,T) into
E, where X^# denotes the space Lipo(X,K) and T is the topology of pointwise
convergence of X^#. The concept of Lipschitz tensor product of elements of X^#
and E^* yields the space X^#\boxast E^* as a certain linear subspace of the
algebraic dual of X\boxtimes E. To ensure the good behavior of a norm on
X\boxtimes E with respect to the Lipschitz tensor product of Lipschitz
functionals (mappings) and bounded linear functionals (operators), the concept
of dualizable (respectively, uniform) Lipschitz cross-norm on X\boxtimes E is
defined. We show that the Lipschitz injective norm epsilon, the Lipschitz
projective norm pi and the Lipschitz p-nuclear norm d_p (1<=p<=infty) are
uniform dualizable Lipschitz cross-norms on X\boxtimes E. In fact, epsilon is
the least dualizable Lipschitz cross-norm and pi is the greatest Lipschitz
cross-norm on X\boxtimes E. Moreover, dualizable Lipschitz cross-norms alpha on
X\boxtimes E are characterized by satisfying the relation epsilon<=alpha<=pi.
In addition, the Lipschitz injective (projective) norm on X\boxtimes E can be
identified with the injective (respectively, projective) tensor norm on the
Banach-space tensor product between the Lipschitz-free space over X and E. In
terms of the space X^#\boxast E^*, we describe the spaces of Lipschitz compact
(finite-rank, approximable) operators from X to E^$.</description><subject>Mathematics - Functional Analysis</subject><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2014</creationdate><recordtype>article</recordtype><sourceid>GOX</sourceid><recordid>eNotzjsPgjAUBeAuDgbdmQx_AGzT20JHQ3wlJC7s5NJHJFEhBY366wV1Omc5Jx8hIaMJZELQNfpn80gY0CxhWQpzEhZN1-tzM7yjwd761kedb81dDwsyc3jp7fKfASl32zI_xMVpf8w3RYxSQOzAUSFQK8DaSTSCaazVWLgCUFxJsAylM1TKjDOulNAaTcpg3FlrOA_I6nf7lVWdb67oX9UkrCYh_wB7ZTU-</recordid><startdate>20140808</startdate><enddate>20140808</enddate><creator>Cabrera-Padilla, M. G</creator><creator>Chávez-Domínguez, J. A</creator><creator>Jiménez-Vargas, A</creator><creator>Villegas-Vallecillos, Moisés</creator><scope>AKZ</scope><scope>GOX</scope></search><sort><creationdate>20140808</creationdate><title>Lipschitz tensor product</title><author>Cabrera-Padilla, M. G ; Chávez-Domínguez, J. A ; Jiménez-Vargas, A ; Villegas-Vallecillos, Moisés</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-a654-f4f055ac94abf6ad51cab96ad394493964e1a6fd0668313995ccad7144f0eed33</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2014</creationdate><topic>Mathematics - Functional Analysis</topic><toplevel>online_resources</toplevel><creatorcontrib>Cabrera-Padilla, M. G</creatorcontrib><creatorcontrib>Chávez-Domínguez, J. A</creatorcontrib><creatorcontrib>Jiménez-Vargas, A</creatorcontrib><creatorcontrib>Villegas-Vallecillos, Moisés</creatorcontrib><collection>arXiv Mathematics</collection><collection>arXiv.org</collection></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext_linktorsrc</fulltext></delivery><addata><au>Cabrera-Padilla, M. G</au><au>Chávez-Domínguez, J. A</au><au>Jiménez-Vargas, A</au><au>Villegas-Vallecillos, Moisés</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Lipschitz tensor product</atitle><date>2014-08-08</date><risdate>2014</risdate><abstract>Inspired by ideas of R. Schatten in his celebrated monograph on a theory of
cross-spaces, we introduce the notion of a Lipschitz tensor product X\boxtimes
E of a pointed metric space and a Banach space E as a certain linear subspace
of the algebraic dual of Lipo(X,E^*). We prove that <Lipo(X,E^*),X\boxtimes E>
forms a dual pair. We prove that X\boxtimes E is linearly isomorphic to the
linear space of all finite-rank continuous linear operators from (X^#,T) into
E, where X^# denotes the space Lipo(X,K) and T is the topology of pointwise
convergence of X^#. The concept of Lipschitz tensor product of elements of X^#
and E^* yields the space X^#\boxast E^* as a certain linear subspace of the
algebraic dual of X\boxtimes E. To ensure the good behavior of a norm on
X\boxtimes E with respect to the Lipschitz tensor product of Lipschitz
functionals (mappings) and bounded linear functionals (operators), the concept
of dualizable (respectively, uniform) Lipschitz cross-norm on X\boxtimes E is
defined. We show that the Lipschitz injective norm epsilon, the Lipschitz
projective norm pi and the Lipschitz p-nuclear norm d_p (1<=p<=infty) are
uniform dualizable Lipschitz cross-norms on X\boxtimes E. In fact, epsilon is
the least dualizable Lipschitz cross-norm and pi is the greatest Lipschitz
cross-norm on X\boxtimes E. Moreover, dualizable Lipschitz cross-norms alpha on
X\boxtimes E are characterized by satisfying the relation epsilon<=alpha<=pi.
In addition, the Lipschitz injective (projective) norm on X\boxtimes E can be
identified with the injective (respectively, projective) tensor norm on the
Banach-space tensor product between the Lipschitz-free space over X and E. In
terms of the space X^#\boxast E^*, we describe the spaces of Lipschitz compact
(finite-rank, approximable) operators from X to E^$.</abstract><doi>10.48550/arxiv.1408.1874</doi><oa>free_for_read</oa></addata></record> |
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| source | arXiv.org |
| subjects | Mathematics - Functional Analysis |
| title | Lipschitz tensor product |
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