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On the approximate fixed point property in abstract spaces
Let X be a Hausdorff topological vector space, X * its topological dual and Z a subset of X * . In this paper, we establish some results concerning the σ ( X , Z )-approximate fixed point property for bounded, closed convex subsets C of X . Three major situations are studied. First, when Z is separa...
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| Published in: | Mathematische Zeitschrift 2012-08, Vol.271 (3-4), p.1271-1285 |
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| Main Authors: | , , |
| Format: | Article |
| Language: | English |
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| cited_by | cdi_FETCH-LOGICAL-c288t-266f22a2d486bc6b267f1273099022eaf938c3514e0abd2344723e48334f28453 |
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| cites | cdi_FETCH-LOGICAL-c288t-266f22a2d486bc6b267f1273099022eaf938c3514e0abd2344723e48334f28453 |
| container_end_page | 1285 |
| container_issue | 3-4 |
| container_start_page | 1271 |
| container_title | Mathematische Zeitschrift |
| container_volume | 271 |
| creator | Barroso, C. S. Kalenda, O. F. K. Lin, P.-K. |
| description | Let
X
be a Hausdorff topological vector space,
X
*
its topological dual and
Z
a subset of
X
*
. In this paper, we establish some results concerning the
σ
(
X
,
Z
)-approximate fixed point property for bounded, closed convex subsets
C
of
X
. Three major situations are studied. First, when
Z
is separable in the strong topology. Second, when
X
is a metrizable locally convex space and
Z
=
X
*
, and third when
X
is not necessarily metrizable but admits a metrizable locally convex topology compatible with the duality. Our approach focuses on establishing the Fréchet–Urysohn property for certain sets with regarding the
σ
(
X
,
Z
)-topology. The support tools include the Brouwer’s fixed point theorem and an analogous version of the classical Rosenthal’s
ℓ
1
-theorem for
ℓ
1
-sequences in metrizable case. The results are novel and generalize previous work obtained by the authors in Banach spaces. |
| doi_str_mv | 10.1007/s00209-011-0915-6 |
| format | article |
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X
be a Hausdorff topological vector space,
X
*
its topological dual and
Z
a subset of
X
*
. In this paper, we establish some results concerning the
σ
(
X
,
Z
)-approximate fixed point property for bounded, closed convex subsets
C
of
X
. Three major situations are studied. First, when
Z
is separable in the strong topology. Second, when
X
is a metrizable locally convex space and
Z
=
X
*
, and third when
X
is not necessarily metrizable but admits a metrizable locally convex topology compatible with the duality. Our approach focuses on establishing the Fréchet–Urysohn property for certain sets with regarding the
σ
(
X
,
Z
)-topology. The support tools include the Brouwer’s fixed point theorem and an analogous version of the classical Rosenthal’s
ℓ
1
-theorem for
ℓ
1
-sequences in metrizable case. The results are novel and generalize previous work obtained by the authors in Banach spaces.</description><identifier>ISSN: 0025-5874</identifier><identifier>EISSN: 1432-1823</identifier><identifier>DOI: 10.1007/s00209-011-0915-6</identifier><language>eng</language><publisher>Berlin/Heidelberg: Springer-Verlag</publisher><subject>Mathematics ; Mathematics and Statistics</subject><ispartof>Mathematische Zeitschrift, 2012-08, Vol.271 (3-4), p.1271-1285</ispartof><rights>Springer-Verlag 2011</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c288t-266f22a2d486bc6b267f1273099022eaf938c3514e0abd2344723e48334f28453</citedby><cites>FETCH-LOGICAL-c288t-266f22a2d486bc6b267f1273099022eaf938c3514e0abd2344723e48334f28453</cites></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><link.rule.ids>314,780,784,27924,27925</link.rule.ids></links><search><creatorcontrib>Barroso, C. S.</creatorcontrib><creatorcontrib>Kalenda, O. F. K.</creatorcontrib><creatorcontrib>Lin, P.-K.</creatorcontrib><title>On the approximate fixed point property in abstract spaces</title><title>Mathematische Zeitschrift</title><addtitle>Math. Z</addtitle><description>Let
X
be a Hausdorff topological vector space,
X
*
its topological dual and
Z
a subset of
X
*
. In this paper, we establish some results concerning the
σ
(
X
,
Z
)-approximate fixed point property for bounded, closed convex subsets
C
of
X
. Three major situations are studied. First, when
Z
is separable in the strong topology. Second, when
X
is a metrizable locally convex space and
Z
=
X
*
, and third when
X
is not necessarily metrizable but admits a metrizable locally convex topology compatible with the duality. Our approach focuses on establishing the Fréchet–Urysohn property for certain sets with regarding the
σ
(
X
,
Z
)-topology. The support tools include the Brouwer’s fixed point theorem and an analogous version of the classical Rosenthal’s
ℓ
1
-theorem for
ℓ
1
-sequences in metrizable case. The results are novel and generalize previous work obtained by the authors in Banach spaces.</description><subject>Mathematics</subject><subject>Mathematics and Statistics</subject><issn>0025-5874</issn><issn>1432-1823</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2012</creationdate><recordtype>article</recordtype><recordid>eNp9j01LxDAQhoMoWFd_gLf8gehkkqapN1n8goW96DmkbaJdtA1JhN1_b5Z63tPAvPMM70PILYc7DtDcJwCElgHnDFpeM3VGKi4FMq5RnJOqxDWrdSMvyVVKO4ASNrIiD9uJ5i9HbQhx3o8_Njvqx70baJjHKdOyDS7mAx0naruUo-0zTcH2Ll2TC2-_k7v5nyvy8fz0vn5lm-3L2_pxw3rUOjNUyiNaHKRWXa86VI3n2AhoW0B01rdC96Lm0oHtBhRSNiic1EJIj1rWYkX48rePc0rReRNiKRoPhoM5yptF3hR5c5Q3qjC4MKncTp8umt38G6dS8wT0B0QOWyw</recordid><startdate>20120801</startdate><enddate>20120801</enddate><creator>Barroso, C. S.</creator><creator>Kalenda, O. F. K.</creator><creator>Lin, P.-K.</creator><general>Springer-Verlag</general><scope>AAYXX</scope><scope>CITATION</scope></search><sort><creationdate>20120801</creationdate><title>On the approximate fixed point property in abstract spaces</title><author>Barroso, C. S. ; Kalenda, O. F. K. ; Lin, P.-K.</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c288t-266f22a2d486bc6b267f1273099022eaf938c3514e0abd2344723e48334f28453</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2012</creationdate><topic>Mathematics</topic><topic>Mathematics and Statistics</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Barroso, C. S.</creatorcontrib><creatorcontrib>Kalenda, O. F. K.</creatorcontrib><creatorcontrib>Lin, P.-K.</creatorcontrib><collection>CrossRef</collection><jtitle>Mathematische Zeitschrift</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Barroso, C. S.</au><au>Kalenda, O. F. K.</au><au>Lin, P.-K.</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>On the approximate fixed point property in abstract spaces</atitle><jtitle>Mathematische Zeitschrift</jtitle><stitle>Math. Z</stitle><date>2012-08-01</date><risdate>2012</risdate><volume>271</volume><issue>3-4</issue><spage>1271</spage><epage>1285</epage><pages>1271-1285</pages><issn>0025-5874</issn><eissn>1432-1823</eissn><abstract>Let
X
be a Hausdorff topological vector space,
X
*
its topological dual and
Z
a subset of
X
*
. In this paper, we establish some results concerning the
σ
(
X
,
Z
)-approximate fixed point property for bounded, closed convex subsets
C
of
X
. Three major situations are studied. First, when
Z
is separable in the strong topology. Second, when
X
is a metrizable locally convex space and
Z
=
X
*
, and third when
X
is not necessarily metrizable but admits a metrizable locally convex topology compatible with the duality. Our approach focuses on establishing the Fréchet–Urysohn property for certain sets with regarding the
σ
(
X
,
Z
)-topology. The support tools include the Brouwer’s fixed point theorem and an analogous version of the classical Rosenthal’s
ℓ
1
-theorem for
ℓ
1
-sequences in metrizable case. The results are novel and generalize previous work obtained by the authors in Banach spaces.</abstract><cop>Berlin/Heidelberg</cop><pub>Springer-Verlag</pub><doi>10.1007/s00209-011-0915-6</doi><tpages>15</tpages></addata></record> |
| fulltext | fulltext |
| identifier | ISSN: 0025-5874 |
| ispartof | Mathematische Zeitschrift, 2012-08, Vol.271 (3-4), p.1271-1285 |
| issn | 0025-5874 1432-1823 |
| language | eng |
| recordid | cdi_crossref_primary_10_1007_s00209_011_0915_6 |
| source | Springer Nature |
| subjects | Mathematics Mathematics and Statistics |
| title | On the approximate fixed point property in abstract spaces |
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