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- Let be a pairing of vector spaces over the field and be a non-empty collection of -bounded subsets of Then,
If covers then the -topology on is Hausdorff.
If distinguishes points of and if is a -dense subset of then the -topology on is Hausdorff.
If is a dual system then the -topology on is Hausdorff if and only if span of is dense in (en)
- Let is a pairing of vector spaces over and let be a non-empty collection of -bounded subsets of The -topology on is not altered if is replaced by any of the following collections of [\sigma-bounded] subsets of :
all subsets of all finite unions of sets in ;
all scalar multiples of all sets in ;
the balanced hull of every set in ;
the convex hull of every set in ;
the -closure of every set in ;
the -closure of the convex balanced hull of every set in (en)
- For any subset the following are equivalent:
is an absorbing subset of
* If this condition is not satisfied then can not possibly be a neighborhood of the origin in any TVS topology on ;
is a -bounded set; said differently, is a bounded subset of ;
for all where this supremum may also be denoted by
The -bounded subsets of have an analogous characterization. (en)
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- Polar topology (en)
- Topologia polare (it)
- 極位相 (ja)
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