# Complementary Projection of Complementary Projection is Projection

## Theorem

Let $\HH$ be a Hilbert space.

Let $I$ be an identity operator on $\HH$.

Let $A$ be a projection.

Let $B$ be the complementary projection of A.

Then the complementary projection of $B$ is $A$.

## Proof

Let $C$ be the complementary projection of $B$.

We have:

 $\ds C$ $=$ $\ds I - B$ Definition of Complementary Projection $\ds$ $=$ $\ds I - \paren{I - A}$ Definition of Complementary Projection $\ds$ $=$ $\ds A$

$\blacksquare$