bounded set

A function taking values in a metric space is bounded if its image is a bounded set.

The underlying idea is the following: a bounded set can be covered by a single ball of some radius.

For example, the left or right order topology on a bounded set provides an example of a compact space that is not Hausdorff.

Furthermore, any bounded set in H is weakly compact.

A bounded set would describe a group with clear "in" and "out" definitions of membership.

The closure of a bounded set is bounded.

The polar of a bounded set is an absolutely convex and absorbing set.

Note that this definition does not require that the domain of the given function is a bounded set.

However, in any case a bounded set can be specified which contains the actual value of D.

Image of any bounded set under T is relatively compact in Y.