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Concept
Absolutely convex set
Summary
In mathematics, a subset C of a real or complex vector space is said to be absolutely convex or disked if it is convex and balanced (some people use the term "circled" instead of "balanced"), in which case it is called a disk.
The disked hull or the absolute convex hull of a set is the intersection of all disks containing that set.
A subset of a real or complex vector space is called a and is said to be , , and if any of the following equivalent conditions is satisfied:
is a convex and balanced set.
for any scalars and if then
for all scalars and if then
for any scalars and if then
for any scalars if then
The smallest convex (respectively, balanced) subset of containing a given set is called the convex hull (respectively, the balanced hull) of that set and is denoted by (respectively, ).
Similarly, the , the , and the of a set is defined to be the smallest disk (with respect to subset inclusion) containing
The disked hull of will be denoted by or and it is equal to each of the following sets:
which is the convex hull of the balanced hull of ; thus,
In general, is possible, even in finite dimensional vector spaces.
the intersection of all disks containing
The intersection of arbitrarily many absolutely convex sets is again absolutely convex; however, unions of absolutely convex sets need not be absolutely convex anymore.
If is a disk in then is absorbing in if and only if
Topological vector space#Properties
If is an absorbing disk in a vector space then there exists an absorbing disk in such that
If is a disk and and are scalars then and
The absolutely convex hull of a bounded set in a locally convex topological vector space is again bounded.
If is a bounded disk in a TVS and if is a sequence in then the partial sums are Cauchy, where for all In particular, if in addition is a sequentially complete subset of then this series converges in to some point of
The convex balanced hull of contains both the convex hull of and the balanced hull of Furthermore, it contains the balanced hull of the convex hull of thus
where the example below shows that this inclusion might be strict.
Official source
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