Complete set
in a topological vector space over a field
A set such that the set of linear combinations of the elements from
is (everywhere) dense in
, i.e. the closed subspace generated by the set
, i.e. the closed linear hull of
, coincides with
. For example, in the normed space
of continuous functions on
with values in
the set
is a complete set. If
is a non-discretely normed field, each absorbing set (and in particular each neighbourhood of zero in
) is a complete set.
In order for ,
, to be a complete set in the weak topology
of a space
it is necessary and sufficient that there exist an index
such that
for each
; this means that no closed hyperplane contains all the elements
, i.e. that
is a total set. Moreover, if
is a locally convex space, a complete set in the weak topology will be a complete set in the initial topology also.
Comments
Of course, a complete set in a topological vector space can also mean a set such that every Cauchy sequence in
converges in
; and this is by far the most frequently occurring meaning of the phrase. For the notion of an absorbing set cf. Topological vector space.
Complete set. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Complete_set&oldid=32720