Characterization of Projections

Theorem
Let $H$ be a Hilbert space.

Let $A \in B \left({H}\right)$ be an idempotent operator.

Then the following are equivalent:


 * $(1): \qquad A$ is a projection
 * $(2): \qquad A$ is the orthogonal projection onto $\operatorname{ran} A$
 * $(3): \qquad \left\Vert{A}\right\Vert = 1$, where $\left\Vert{\cdot}\right\Vert$ is the norm on bounded linear operators.
 * $(4): \qquad A$ is self-adjoint
 * $(5): \qquad A$ is normal
 * $(6): \qquad \forall h \in H: \left\langle{Ah, h}\right\rangle_H \ge 0$