Namespaces
Variants
Actions

Difference between revisions of "Free set"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
m (tex encoded by computer)
 
Line 1: Line 1:
''in a vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041650/f0416501.png" /> over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041650/f0416502.png" />''
+
<!--
 +
f0416501.png
 +
$#A+1 = 38 n = 0
 +
$#C+1 = 38 : ~/encyclopedia/old_files/data/F041/F.0401650 Free set
 +
Automatically converted into TeX, above some diagnostics.
 +
Please remove this comment and the {{TEX|auto}} line below,
 +
if TeX found to be correct.
 +
-->
  
A linearly independent system of vectors from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041650/f0416503.png" />, that is, a set of elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041650/f0416504.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041650/f0416505.png" />, such that the relation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041650/f0416506.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041650/f0416507.png" /> for all but a finite number of indices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041650/f0416508.png" />, implies that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041650/f0416509.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041650/f04165010.png" />. A non-free set is also called dependent.
+
{{TEX|auto}}
 +
{{TEX|done}}
  
A free set in a topological vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041650/f04165011.png" /> over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041650/f04165012.png" /> (a topologically-free set) is a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041650/f04165013.png" /> such that for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041650/f04165014.png" /> the closed subspace generated by the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041650/f04165015.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041650/f04165016.png" />, does not contain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041650/f04165017.png" />. A topologically-free set is a free set in the vector space; the converse is not true. For example, in the normed space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041650/f04165018.png" /> of continuous functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041650/f04165019.png" />, the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041650/f04165020.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041650/f04165021.png" />, form a topologically-free set, in contrast to the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041650/f04165022.png" /> (since, e.g., <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041650/f04165023.png" /> is contained in the closed subspace generated by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041650/f04165024.png" />).
+
''in a vector space $  X $
 +
over a field $  K $''
  
The set of all (topologically-) free sets in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041650/f04165025.png" /> is, in general, not inductive under inclusion; in addition, it does not necessarily contain a maximal topologically-free set. For example, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041650/f04165026.png" /> be the space over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041650/f04165027.png" /> formed by the continuous functions and endowed with the following Hausdorff topology: a fundamental system of neighbourhoods of zero in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041650/f04165028.png" /> consists of the balanced absorbing sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041650/f04165029.png" />. Then every continuous linear functional vanishes, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041650/f04165030.png" /> does not contain a maximal free set.
+
A linearly independent system of vectors from  $  X $,
 +
that is, a set of elements  $  A = \{ a _ {t} \} \subset  X $,  
 +
$  t \in T $,  
 +
such that the relation  $  \sum \xi _ {t} a _ {t} = 0 $,
 +
where  $  \xi _ {t} = 0 $
 +
for all but a finite number of indices  $  t $,  
 +
implies that  $  \xi _ {t} = 0 $
 +
for all  $  t $.  
 +
A non-free set is also called dependent.
  
For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041650/f04165031.png" /> to be a (topologically-) free set in the weak topology <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041650/f04165032.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041650/f04165033.png" /> it is necessary and sufficient that for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041650/f04165034.png" /> there is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041650/f04165035.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041650/f04165036.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041650/f04165037.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041650/f04165038.png" />. For a locally convex space a free set in the weak topology is a free set in the original topology.
+
A free set in a topological vector space  $  X $
 +
over a field  $  K $(
 +
a topologically-free set) is a set  $  A = \{ a _ {t} \} \subset  X $
 +
such that for any  $  s \in T $
 +
the closed subspace generated by the points  $  a _ {t} $,
 +
$  t \neq s $,
 +
does not contain  $  a _ {s} $.
 +
A topologically-free set is a free set in the vector space; the converse is not true. For example, in the normed space  $  C $
 +
of continuous functions on  $  [ 0, 1] $,
 +
the functions  $  \mathop{\rm exp} [ 2 \pi kix] $,
 +
$  k \in Z $,
 +
form a topologically-free set, in contrast to the functions  $  x  ^ {k} $(
 +
since, e.g.,  $  x $
 +
is contained in the closed subspace generated by  $  \{ x  ^ {2k} \} $).
 +
 
 +
The set of all (topologically-) free sets in  $  X $
 +
is, in general, not inductive under inclusion; in addition, it does not necessarily contain a maximal topologically-free set. For example, let  $  X $
 +
be the space over  $  \mathbf R $
 +
formed by the continuous functions and endowed with the following Hausdorff topology: a fundamental system of neighbourhoods of zero in  $  X $
 +
consists of the balanced absorbing sets  $  V _ {s, \epsilon }  = \{ {x } : {| f ( x) | \leq  \delta  \textrm{ everywhere  outside  an  open  set  }  \textrm{ (depending  on  }  f  \textrm{ )  of  measure  }  \leq  \epsilon,  0 \langle  \epsilon < 1,  \delta \rangle 0 } \} $.  
 +
Then every continuous linear functional vanishes, and  $  X $
 +
does not contain a maximal free set.
 +
 
 +
For  $  A $
 +
to be a (topologically-) free set in the weak topology $  \sigma ( X, X  ^ {*} ) $
 +
in $  X $
 +
it is necessary and sufficient that for each $  t $
 +
there is a $  b _ {t} \in X  ^ {*} $
 +
such that $  \langle  a _ {t} , b _ {t} \rangle \neq 0 $,  
 +
and $  \langle  a _ {s} , b _ {t} \rangle = 0 $
 +
for all $  s \neq t $.  
 +
For a locally convex space a free set in the weak topology is a free set in the original topology.

Latest revision as of 19:40, 5 June 2020


in a vector space $ X $ over a field $ K $

A linearly independent system of vectors from $ X $, that is, a set of elements $ A = \{ a _ {t} \} \subset X $, $ t \in T $, such that the relation $ \sum \xi _ {t} a _ {t} = 0 $, where $ \xi _ {t} = 0 $ for all but a finite number of indices $ t $, implies that $ \xi _ {t} = 0 $ for all $ t $. A non-free set is also called dependent.

A free set in a topological vector space $ X $ over a field $ K $( a topologically-free set) is a set $ A = \{ a _ {t} \} \subset X $ such that for any $ s \in T $ the closed subspace generated by the points $ a _ {t} $, $ t \neq s $, does not contain $ a _ {s} $. A topologically-free set is a free set in the vector space; the converse is not true. For example, in the normed space $ C $ of continuous functions on $ [ 0, 1] $, the functions $ \mathop{\rm exp} [ 2 \pi kix] $, $ k \in Z $, form a topologically-free set, in contrast to the functions $ x ^ {k} $( since, e.g., $ x $ is contained in the closed subspace generated by $ \{ x ^ {2k} \} $).

The set of all (topologically-) free sets in $ X $ is, in general, not inductive under inclusion; in addition, it does not necessarily contain a maximal topologically-free set. For example, let $ X $ be the space over $ \mathbf R $ formed by the continuous functions and endowed with the following Hausdorff topology: a fundamental system of neighbourhoods of zero in $ X $ consists of the balanced absorbing sets $ V _ {s, \epsilon } = \{ {x } : {| f ( x) | \leq \delta \textrm{ everywhere outside an open set } \textrm{ (depending on } f \textrm{ ) of measure } \leq \epsilon, 0 \langle \epsilon < 1, \delta \rangle 0 } \} $. Then every continuous linear functional vanishes, and $ X $ does not contain a maximal free set.

For $ A $ to be a (topologically-) free set in the weak topology $ \sigma ( X, X ^ {*} ) $ in $ X $ it is necessary and sufficient that for each $ t $ there is a $ b _ {t} \in X ^ {*} $ such that $ \langle a _ {t} , b _ {t} \rangle \neq 0 $, and $ \langle a _ {s} , b _ {t} \rangle = 0 $ for all $ s \neq t $. For a locally convex space a free set in the weak topology is a free set in the original topology.

How to Cite This Entry:
Free set. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Free_set&oldid=46987
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article