View source for Total set

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, [[#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.

====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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