Balanced set

As a balanced amount in the functional analysis a subset of a real or complex vector space called if for each vector in each number and the vector is also in. So the distance from to is. Balanced amounts of many authors also circular (English circled ), disc-shaped or balanced ( engl. balanced) called.

Is balanced and is not empty, the zero vector must contain, as is, then.

In a topological vector space every neighborhood of the root also contains a balanced neighborhood of zero. Indeed, if a neighborhood of zero, so there is a view of the continuity of the scalar multiplication and a neighborhood of zero, such that for all and all. Then is a balanced neighborhood of zero contained in.

In a topological vector space so there is always a base of neighborhoods of balanced quantities. If one has reversed to an algebraic vector space a system of absorbing and balanced amounts with the properties

• For all true,
• With two volumes, contains the average,
• For each there is a with,
• ,

So is the vector space with a base of neighborhoods of a topological vector space. The balance is required to demonstrate the continuity of the scalar multiplication.

Balanced convex sets are also called absolutely convex. They play an important role in the theory of locally convex spaces.