“Absolutely convex set”的意思、由来-开放百科全书

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Absolutely convex set

释义

Properties
Absolutely convex hull
See also
References

A set C in a real or complex vector space is said to be absolutely convex or disked if it is convex and balanced (circled), in which case it is called a disk.

Properties

A set is absolutely convex if and only if for any points in and any numbers satisfying

the sum

belongs to .

The intersection of arbitrarily many absolutely convex sets is again absolutely convex; however, unions of absolutely convex sets need not be absolutely convex anymore.

Absolutely convex hull

Since the intersection of any collection of absolutely convex sets is absolutely convex, one can define

for any subset A of a vector space its absolutely convex hull as the intersection of all absolutely convex sets containing A, analogous to the well-known construction of the convex hull.

More explicitly, one can define the absolutely convex hull of the set A via

where the λ_i are elements of the underlying field.

The absolutely convex hull of a bounded set in a topological vector space is again bounded.

References

{{cite book |last=Robertson |first=A.P. |author2= W.J. Robertson |title= Topological vector spaces |series=Cambridge Tracts in Mathematics |volume=53 |year=1964 |publisher= Cambridge University Press | pages=4–6 }}
{{cite book

|last= Narici
|first= Lawrence
|last2= Beckenstein
|first2= Edward
|edition= Second
|publisher= Chapman and Hall/CRC
|date= July 26, 2010
|series= Pure and Applied Mathematics
|number= 296
|title= Topological Vector Spaces, Second Edition
}}

{{cite book |last=Schaefer |first=H.H. |title= Topological vector spaces |year=1999 |publisher= Springer-Verlag Press | pages=39 }}

4 : Abstract algebra|Linear algebra|Group theory|Convex geometry

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Properties

Absolutely convex hull

See also

References