Orthogonal Projection is Projection

Theorem

Let $H$ be a Hilbert space.

Let $K$ be a closed linear subspace of $H$.

Let $P_K$ denote the orthogonal projection on $K$.

Then $P_K$ is idempotent, i.e.:

$P_K \circ P_K = P_K$

Proof

Let $h \in H$.

From the definition of the orthogonal projection, we have:

$\map {P_K} h \in K$

So, from Fixed Points of Orthogonal Projection, we have:

$\map {\paren {P_K \circ P_K} } h = \map {P_K} {\map {P_K} h} = \map {P_K} h$

Since $h$ was arbitrary, we have:

$P_K \circ P_K = P_K$

$\blacksquare$

Sources

• 1990: John B. Conway: A Course in Functional Analysis (2nd ed.) ... (previous) ... (next): $\text{I}$ Hilbert Spaces: $\S 2.$ Orthogonality: Theorem $2.7 \text{(c)}$