Difference between revisions of "Total set"
From Encyclopedia of Mathematics
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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 $M \subset T$, 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. | + | 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==== | ||
Latest revision as of 19:44, 27 February 2021
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://encyclopediaofmath.org/index.php?title=Total_set&oldid=39955
Total set. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Total_set&oldid=39955
This article was adapted from an original article by V.I. Lomonosov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article