A projection (or subspace belonging to "M") "E" is said to be a finite projection if there is no projection "F" < "E" (meaning "F" ≤ "E" and "F" ≠ "E") that is equivalent to "E". For example, all finite-dimensional projections (or subspaces) are finite (since isometries between Hilbert spaces leave the dimension fixed), but the identity operator on an infinite-dimensional Hilbert space is not finite in the von Neumann algebra of all bounded operators on it, since it is isometrically isomorphic to a proper subset of itself. However it is possible for infinite dimensional subspaces to be finite.