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Concept
Balanced set
Summary
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.
Official source
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Ontological neighbourhood