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A set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093390/t0933901.png" /> of linear functionals on a [[Vector space|vector space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093390/t0933902.png" /> separating the points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093390/t0933903.png" />, that is, such that for any non-zero vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093390/t0933904.png" /> there is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093390/t0933905.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093390/t0933906.png" />.
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A set $\Sigma$ of linear functionals on a [[vector space]] $E$ separating the points of $E$, that is, such that for any non-zero vector $x \in E$ there is an $f \in \Sigma$ with $f(x) \neq 0$.
  
  
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A total set in the sense above is also, and more precisely, called a total set of linear functions, [[#References|[a1]]].
 
A total set in the sense above is also, and more precisely, called a total set of linear functions, [[#References|[a1]]].
  
More generally, a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093390/t0933907.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093390/t0933908.png" /> is a topological vector space, is called a total set or fundamental set if the linear span of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093390/t0933909.png" /> is dense in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093390/t09339010.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093390/t09339011.png" />, the algebraic dual of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093390/t09339012.png" />, is given the weak topology (so that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093390/t09339013.png" />, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093390/t09339014.png" /> is the base field and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093390/t09339015.png" /> is an (algebraic) basis for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093390/t09339016.png" />), the two definitions for a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093390/t09339017.png" /> agree.
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More generally, a set $M \subset T$, where $T$ is a topological vector space, is called a total set or fundamental set if the linear span of $M$ is dense in $T$. If the algebraic dual $E^*$ of $E$, is given the weak topology (so that $E^* \simeq \prod_{\alpha \in A} K$,where $K$ is the base field and $\{ e_\alpha : \alpha \in A \}$ is an (algebraic) basis for $E$), the two definitions for a set $\Sigma \subset E^*$ agree.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  S. Rolewicz,  "Metric linear spaces" , Reidel  (1985)  pp. 44</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  G. Köthe,  "Topological vector spaces" , '''1''' , Springer  (1969)  pp. 132, 247ff</TD></TR></table>
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<table>
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<TR><TD valign="top">[a1]</TD> <TD valign="top">  S. Rolewicz,  "Metric linear spaces" , Reidel  (1985)  pp. 44</TD></TR>
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<TR><TD valign="top">[a2]</TD> <TD valign="top">  G. Köthe,  "Topological vector spaces" , '''1''' , Springer  (1969)  pp. 132, 247ff</TD></TR>
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</table>
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{{TEX|done}}

Revision as of 00:04, 11 December 2016

A set $\Sigma$ of linear functionals on a vector space $E$ separating the points of $E$, that is, such that for any non-zero vector $x \in E$ there is an $f \in \Sigma$ with $f(x) \neq 0$.


Comments

A total set in the sense above is also, and more precisely, called a total set of linear functions, [a1].

More generally, a set $M \subset T$, where $T$ is a topological vector space, is called a total set or fundamental set if the linear span of $M$ is dense in $T$. If the algebraic dual $E^*$ of $E$, is given the weak topology (so that $E^* \simeq \prod_{\alpha \in A} K$,where $K$ is the base field and $\{ e_\alpha : \alpha \in A \}$ is an (algebraic) basis for $E$), the two definitions for a set $\Sigma \subset E^*$ agree.

References

[a1] S. Rolewicz, "Metric linear spaces" , Reidel (1985) pp. 44
[a2] G. Köthe, "Topological vector spaces" , 1 , Springer (1969) pp. 132, 247ff
How to Cite This Entry:
Total set. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Total_set&oldid=14064
This article was adapted from an original article by V.I. Lomonosov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article