In linear algebra and related areas of mathematics a balanced set, circled set or disk in a vector space (over a field K with an absolute value |.|) is a set S so that for all scalars α with |α| ≤ 1 Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (or linear spaces), linear transformations, and systems of linear equations. ... Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations by or about: Mathematics Look up Mathematics on Wiktionary, the free dictionary Wikimedia Commons has more media related to: Mathematics Bogomolny, Alexander: Interactive Mathematics Miscellany and Puzzles. ... A vector space (or linear space) is the basic object of study in the branch of mathematics called linear algebra. ... Look up field in Wiktionary, the free dictionary A green field or paddock Field may refer to: A field is an open land area, used for growing agricultural crops. ... In mathematics, the absolute value (or modulus1) of a real number is its numerical value without regard to its sign. ... In mathematics, a set can be thought of as any collection of distinct things considered as a whole. ... Scalar is a concept that has meaning in mathematics, physics, and computing. ...

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The balanced hull or balanced envelope for a set S is the smallest balanced set containing S. It can be constructed as the intersection of all balanced sets containing S. In mathematics, the intersection of two sets A and B is the set that contains all elements of A that also belong to B (or equivalently, all elements of B that also belong to A), but no other elements. ...

Examples

• The unit ball in a normed vector space is a balanced set.
• The vector space V as a whole and the trivial subspace { o__V } are balanced sets.
• The product set of a family of balanced sets is balanced in the product space of the corresponding vector spaces (over the same field K).

some unit spheres In mathematics, a unit sphere is the set of points of distance 1 from a fixed central point, where a generalized concept of distance may be used. ... In mathematics, with 2- or 3-dimensional vectors with real-valued entries, the idea of the length of a vector is intuitive and can be easily extended to any real vector space Rn. ... In topology, the cartesian product of topological spaces is turned into a topological space in the following way. ...

Properties

• The union and intersection of balanced sets is a balanced set.
• By definition, a set is absolutely convex if and only if it is convex and balanced.