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) is a set for all scalars

The balanced hull or balanced envelope The balanced core

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.

Definition

be a vector space over the field of real or complex numbers.

Notation

denote, respectively, the open ball and the closed ball

Balanced set

is called a or balanced if it satisfies any of the following equivalent conditions: Definition ).

• is a balanced set (according to any of the previous defining conditions).

is a balanced set (according to any defining condition other than this one). (according to any defining condition of "balanced" other than this one).

• is balanced" without mentioning any vector space.

is a convex set then this list may be extended to include:

then this list may be extended to include: is symmetric

Balanced hull

The is defined in any of the following equivalent ways: Definition is the intersection

Balanced core

The is defined in any of the following equivalent ways: Definition

Examples

The empty set is a balanced set. As is any vector subspace of any (real or complex) vector space is always a balanced set.

Any non-empty set that does not contain the origin is not balanced and furthermore, the balanced core of such a set will equal the empty set.

Normed and topological vector spaces

The open and closed balls centered at the origin in a normed vector space is a seminorm (or norm is balanced.

is any balanced neighborhood of the origin in a topological vector space

be the field real numbers or complex numbers denote the absolute value is a 1-dimensional real vector space.

are exactly the following: Consequently, both the balanced core and the balanced hull of every set of scalars is equal to one of the sets listed above.

are entirely different as far as scalar multiplication is concerned.

centered at the origin.

is

is the entire vector space).

is a balanced and absorbing set but it is not necessarily convex.

The balanced hull

The next example shows that the balanced hull is a non-convex subset that is "hour glass shaped" and equal to the union of two closed and filled isosceles triangles is the convex hull).

Sufficient conditions

The Cartesian product of a family of balanced sets is balanced in the product space).

The balanced hull of a compact (respectively, totally bounded, bounded) set has the same property. The convex hull of a balanced set is convex and balanced (that is, it is absolutely convex). However, the balanced hull of a convex set may fail to be convex (a counter-example is given above). Arbitrary unions of balanced sets are balanced, and the same is true of arbitrary intersections of balanced sets. Scalar multiples and (finite) Minkowski sums of balanced sets are again balanced. Images and preimages of balanced sets under linear maps are balanced sets.

Balanced neighborhoods

In any topological vector space, the closure of a balanced set is balanced. and the topological interior of a balanced set is balanced. Therefore, the topological interior of a balanced neighborhood of the origin is balanced. but whose (nonempty) topological interior does not contain the origin and is therefore not a balanced set. (a filled triangle is an (hour glass formed by adding the origin is balanced, it is neither an open set nor a neighborhood of the origin).

Every neighborhood (respectively, convex neighborhood) of the origin in a topological vector space contains a balanced (respectively, convex and balanced) open neighborhood of the origin. the symmetric set will be convex (respectively, closed, balanced, bounded, a neighborhood of the origin, an absorbing subset is a star shaped at the origin, will be a convex neighborhood of the origin and so its topological interior will be a balanced convex open neighborhood of the origin.

is a convex and absorbing subset will be convex balanced which guarantees that the Minkowski functional will be a seminorm into a seminormed space that carries its canonical pseduometrizable as a limit point) forms a neighborhood basis of absorbing disks at the origin for this locally convex is a topological vector space is also a bounded subset will be a norm will form what is known as an auxiliary normed space. If this normed space is a Banach space is called a .

Properties

Properties of balanced sets

A balanced set is not empty if and only if it contains the origin. By definition, a set is absolutely convex if and only if it is convex and balanced. Every balanced set is star-shaped (at 0) and a symmetric set then: is absorbing or else it is absorbing when this space is endowed with the Hausdorff Euclidean topology that contains the origin.

Properties of balanced hulls and balanced cores

In any topological vector space, the balanced hull is a Hausdorff topological vector space is compact.

If a set is closed (respectively, convex, absorbing, a neighborhood of the origin) then the same is true of its balanced core.

Related notions

on a real or complex vector space is said to be a if it satisfies any of the following equivalent conditions: for every unit length shows that every balanced function is a symmetric function.

is a seminorm if and only if it is a balanced sublinear function.

See also

References

Proofs

Sources