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balanced set
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(Definition)
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Definition [1,2,3,4] Let  be a vector space over
 (or
 ), and let  be a subset of  . If
 for all scalars  such that
 , then  is a balanced set in  . The balanced hull of  , denoted by
 , is the smallest balanced set containing  .
In the above,
 , and  is the absolute value (in
 ), or the modulus of a complex number (in
 ).
- Let
 be a normed space with norm  . Then the unit ball
 is a balanced set.
- Any vector subspace is a balanced set. Thus, in
 , lines and planes passing through the origin are balanced sets.
A balanced set is also sometimes called circled [3]. The term balanced evelope is also used for the balanced hull [2]. Bourbaki uses the term équilibré [2], c.f.
 above. In [5], a balanced set is defined as above, but with the condition
 instead of
 .
- 1
- W. Rudin, Functional Analysis, McGraw-Hill Book Company, 1973.
- 2
- R.E. Edwards, Functional Analysis: Theory and Applications, Dover Publications, 1995.
- 3
- J. Horváth, Topological Vector Spaces and Distributions, Addison-Wsley Publishing Company, 1966.
- 4
- R. Cristescu, Topological vector spaces, Noordhoff International Publishing, 1977.
- 5
- M. Reed, B. Simon, Methods of Modern Mathematical Physics: Functional Analysis I, Revised and enlarged edition, Academic Press, 1980.
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"balanced set" is owned by matte. [ full author list (2) ]
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(view preamble)
See Also: absorbing set
| Also defines: |
balanced subset, balanced hull, balanced evelope, circled, équilibré |
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Cross-references: Bourbaki, origin, passing through, planes, lines, vector subspace, unit ball, norm, normed space, modulus of a complex number, absolute value, scalars, subset, vector space
There are 4 references to this entry.
This is version 2 of balanced set, born on 2005-10-28, modified 2005-10-29.
Object id is 7453, canonical name is BalancedSet.
Accessed 4424 times total.
Classification:
| AMS MSC: | 46-00 (Functional analysis :: General reference works ) |
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Pending Errata and Addenda
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