Thread: Balanced and absorbant sets

Hi all,

I am working from the following definitions:

Let V be a vector space over a field F. We say that a set C subset V is balanced if for all x in C, ax is in C if |a| <= 1 for a in F. We say that a set C subset V is absorbent if for every x in V; tx is in C for some t > 0.

I am trying to describe these properties geometrically, and I've not found a source that does this. My interepretations are:

Balanced: the set has no isolated points, there are always lots of points around it, it seems a bit like completeness, but not in the usual sense.

Absorbent: A bit like a basis, this set can generate the whole vector space by scalar multiplication.


Does anyone have a better idea?

Thanks.

Originally Posted by measureman

Hi all,

I am working from the following definitions:

Let V be a vector space over a field F. We say that a set C subset V is balanced if for all x in C, ax is in C if |a| <= 1 for a in F. We say that a set C subset V is absorbent if for every x in V; tx is in C for some t > 0.

I am trying to describe these properties geometrically, and I've not found a source that does this. My interepretations are:

Balanced: the set has no isolated points, there are always lots of points around it, it seems a bit like completeness, but not in the usual sense.

Absorbent: A bit like a basis, this set can generate the whole vector space by scalar multiplication.


Does anyone have a better idea?

Thanks.

For the definition of balanced to make sense, the field F must be equipped with an absolute value. If for example F is the real field, then the set C is balanced if, for each x in C, the entire line segment between x and –x lies in C. That implies that the origin lies in C, and that C is "balanced" about the origin in the sense that it is symmetric with respect to reflection through the origin.

If the scalar field is the complex numbers, then the definition of balanced becomes stronger, and not so easy to visualise geometrically.

An absorbent set is, as you say, one that can generate, or "absorb", the whole vector space by scalar multiplication. If the space V is a topological vector space, then any neighbourhood of the origin will be absorbent. You can think of the concept of being absorbent as a device for giving a topological-type property to a space that does not have a topology.