Polar topology
http://dbpedia.org/resource/Polar_topology
In functional analysis and related areas of mathematics a polar topology, topology of -convergence or topology of uniform convergence on the sets of is a method to define locally convex topologies on the vector spaces of a pairing.
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数学の関数解析学の分野における極位相（きょくいそう、英: polar topology）あるいは-収束の位相またはの集合上の一様収束位相とは、双対組のベクトル空間に対して定義されるある局所凸位相のことをいう。
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In matematica, in particolare in analisi funzionale, una topologia polare consente di definire una topologia localmente convessa su una coppia di spazi vettoriali duali (in generale relazionati mediante una forma bilineare).
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Topologia polare
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極位相
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Polar topology
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Theorem
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1789978
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1105098167
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true
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Proof
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In functional analysis and related areas of mathematics a polar topology, topology of -convergence or topology of uniform convergence on the sets of is a method to define locally convex topologies on the vector spaces of a pairing.
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数学の関数解析学の分野における極位相（きょくいそう、英: polar topology）あるいは-収束の位相またはの集合上の一様収束位相とは、双対組のベクトル空間に対して定義されるある局所凸位相のことをいう。
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In matematica, in particolare in analisi funzionale, una topologia polare consente di definire una topologia localmente convessa su una coppia di spazi vettoriali duali (in generale relazionati mediante una forma bilineare).
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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
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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
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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.
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43807