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Difference between revisions of "Locally convex lattice"

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A real [[Topological vector space|topological vector space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060350/l0603501.png" /> that is simultaneously a [[Vector lattice|vector lattice]] and whose topology is a [[Locally convex topology|locally convex topology]], while the mappings of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060350/l0603502.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060350/l0603503.png" /> defined by
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A real [[topological vector space]] $E$ that is simultaneously a [[vector lattice]] and whose topology is a [[locally convex topology]], while the mappings of $E \times E$ into $E$ defined by
 
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$$
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060350/l0603504.png" /></td> </tr></table>
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(x,y) \mapsto \sup\{x,y\} \,,\ \ \ (x,y) \mapsto \inf\{x,y\} \ \ \text{for}\ x,y \in E \,,
 
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$$
are continuous. General questions in the theory of locally convex lattices are the following: The study of the connections between topological properties and order properties; in particular, the topological properties of bands and positive cones in a locally convex lattice and connections between lattice properties and topological properties of completeness in a locally convex lattice. The study of properties of the strong dual of a locally convex lattice and properties of the imbedding of a locally convex lattice <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060350/l0603505.png" /> into its second dual. The construction of a theory of extension of positive functionals and linear mappings between locally convex lattices.
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are continuous. General questions in the theory of locally convex lattices are the following: The study of the connections between topological properties and order properties; in particular, the topological properties of bands and positive cones in a locally convex lattice and connections between lattice properties and topological properties of completeness in a locally convex lattice. The study of properties of the strong dual of a locally convex lattice and properties of the imbedding of a locally convex lattice $E$ into its second dual. The construction of a theory of extension of positive functionals and linear mappings between locally convex lattices.
  
 
The most important example of a locally convex lattice is a [[Banach lattice|Banach lattice]].
 
The most important example of a locally convex lattice is a [[Banach lattice|Banach lattice]].
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  L.V. Kantorovich,  G.P. Akilov,  "Functional analysis" , Pergamon  (1982)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  M.M. Day,  "Normed linear spaces" , Springer  (1958)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  H.H. Schaefer,  "Topological vector spaces" , Macmillan  (1966)</TD></TR></table>
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<table>
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<TR><TD valign="top">[1]</TD> <TD valign="top">  L.V. Kantorovich,  G.P. Akilov,  "Functional analysis" , Pergamon  (1982)  (Translated from Russian)</TD></TR>
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<TR><TD valign="top">[2]</TD> <TD valign="top">  M.M. Day,  "Normed linear spaces" , Springer  (1958)</TD></TR>
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<TR><TD valign="top">[3]</TD> <TD valign="top">  H.H. Schaefer,  "Topological vector spaces" , Macmillan  (1966)</TD></TR>
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</table>
  
  
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====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  W.A.J. Luxemburg,  A.C. Zaanen,  "Riesz spaces" , '''I''' , North-Holland  (1971)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  A.C. Zaanen,  "Riesz spaces" , '''II''' , North-Holland  (1983)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  H.H. Schaefer,  "Banach lattices and positive operators" , Springer  (1974)</TD></TR></table>
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<table>
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<TR><TD valign="top">[a1]</TD> <TD valign="top">  W.A.J. Luxemburg,  A.C. Zaanen,  "Riesz spaces" , '''I''' , North-Holland  (1971)</TD></TR>
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<TR><TD valign="top">[a2]</TD> <TD valign="top">  A.C. Zaanen,  "Riesz spaces" , '''II''' , North-Holland  (1983)</TD></TR>
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<TR><TD valign="top">[a3]</TD> <TD valign="top">  H.H. Schaefer,  "Banach lattices and positive operators" , Springer  (1974)</TD></TR>
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</table>
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Latest revision as of 16:48, 14 October 2016

A real topological vector space $E$ that is simultaneously a vector lattice and whose topology is a locally convex topology, while the mappings of $E \times E$ into $E$ defined by $$ (x,y) \mapsto \sup\{x,y\} \,,\ \ \ (x,y) \mapsto \inf\{x,y\} \ \ \text{for}\ x,y \in E \,, $$ are continuous. General questions in the theory of locally convex lattices are the following: The study of the connections between topological properties and order properties; in particular, the topological properties of bands and positive cones in a locally convex lattice and connections between lattice properties and topological properties of completeness in a locally convex lattice. The study of properties of the strong dual of a locally convex lattice and properties of the imbedding of a locally convex lattice $E$ into its second dual. The construction of a theory of extension of positive functionals and linear mappings between locally convex lattices.

The most important example of a locally convex lattice is a Banach lattice.

References

[1] L.V. Kantorovich, G.P. Akilov, "Functional analysis" , Pergamon (1982) (Translated from Russian)
[2] M.M. Day, "Normed linear spaces" , Springer (1958)
[3] H.H. Schaefer, "Topological vector spaces" , Macmillan (1966)


Comments

References

[a1] W.A.J. Luxemburg, A.C. Zaanen, "Riesz spaces" , I , North-Holland (1971)
[a2] A.C. Zaanen, "Riesz spaces" , II , North-Holland (1983)
[a3] H.H. Schaefer, "Banach lattices and positive operators" , Springer (1974)
How to Cite This Entry:
Locally convex lattice. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Locally_convex_lattice&oldid=39415
This article was adapted from an original article by A.I. Shtern (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article