# Free set

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