Transformations are modeled as functions acting on the entire space, meaning that every transformation must be applicable to every object.

In physics, an operator is a function acting on the space of physical states.

Now consider acting on the space of all forms .

In the Schrödinger model, the Heisenberg group acts on the space of square integrable functions.

In contrast, no such forces act on the space between objects.

The induced representation can be thought of as acting on the following space:

To explain how the pentagram map acts on the moduli space, one must say a few words about the torus.

All of these models have the conceptual problem of requiring an outside force acting on the "space" at all times to make it expand.

Then, also acts on the space of holomorphic functions from to the complex numbers.

Therefore the Poincaré group also acts on the space of sections.