Orthogonal Projection is Projection

Theorem

Let $\HH$ be a Hilbert space.

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

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


Then $P_K$ is a projection.


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:

$\map {P_K^2} h = \map {P_K} {\map {P_K} h} = \map {P_K} h$

Since $h$ was arbitrary, we have:

$P_K^2 = P_K$

So $P_K$ is an idempotent.

Further, from Kernel of Orthogonal Projection, we have:

$\ker {P_K} = K^\bot$

where $K^\bot$ is the orthocomplement of $K$.

From Range of Orthogonal Projection, we have:

$K = \Img {P_K}$

Hence we conclude:

$\ker {P_K} = \paren {\Img {P_K} }^\bot$

Hence $P_K$ is a projection.

$\blacksquare$


Sources

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