Balanced set

In linear algebra and related areas of mathematics a balanced set, circled set or disk in a vector space (over a field with an absolute value function ) is a set such that for all scalars satisfying The balanced hull or balanced envelope of a set is the smallest balanced set containing The balanced core of a set is the largest balanced set contained in Balanced sets are ubiquitous in functional analysis because every neighborhood of the origin in every topological vector space (TVS) contains a balanced neighborhood of the origin and every convex neighborhood of the origin contains a balanced convex neighborhood of the origin (even if the TVS is not locally convex). This neighborhood can also be chosen to be an open set or, alternatively, a closed set. Let be a vector space over the field of real or complex numbers. Notation If is a set, is a scalar, and then let and and for any let denote, respectively, the open ball and the closed ball of radius in the scalar field centered at where and Every balanced subset of the field is of the form or for some Balanced set A subset of is called a or balanced if it satisfies any of the following equivalent conditions: Definition: for all and all scalars satisfying for all scalars satisfying where For every is a (if ) or (if ) dimensional vector subspace of If then the above equality becomes which is exactly the previous condition for a set to be balanced. Thus, is balanced if and only if for every is a balanced set (according to any of the previous defining conditions). For every 1-dimensional vector subspace of is a balanced set (according to any defining condition other than this one). For every there exists some such that or If is a convex set then this list may be extended to include: for all scalars satisfying If then this list may be extended to include: is symmetric (meaning ) and The of a subset of denoted by is defined in any of the following equivalent ways: Definition: is the smallest (with respect to ) balanced subset of containing is the intersection of all balanced sets containing The of a subset of denoted by is defined in any of the following equivalent ways: Definition: is the largest (with respect to ) balanced subset of is the union of all balanced subsets of if while if The empty set is a balanced set.