Characterization of Projections

Theorem
Let $H$ be a Hilbert space.

Let $A \in \map B H$ be an idempotent operator.

Then the following are equivalent:


 * $(1): \qquad A$ is a projection
 * $(2): \qquad A$ is the orthogonal projection onto $\Rng A$
 * $(3): \qquad \norm A = 1$, where $\norm {\, \cdot \,}$ is the norm on bounded linear operators.
 * $(4): \qquad A$ is self-adjoint
 * $(5): \qquad A$ is normal
 * $(6): \qquad \forall h \in H: \innerprod {A h} h_H \ge 0$