Properties of Orthogonal Projection
Theorem
Orthogonal Projection is Linear Transformation
Let $\GF \in \set {\R, \C}$.
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Let $H$ be a Hilbert space over $\mathbb F$ with inner product $\innerprod \cdot \cdot$.
Let $K$ be a closed linear subspace of $H$.
Let $P_K$ denote the orthogonal projection on $K$.
Then $P_K$ is a linear transformation on $H$.
Orthogonal Projection is Bounded
Let $H$ be a Hilbert space with inner product $\innerprod \cdot \cdot$ and inner product norm $\norm \cdot$.
Let $K$ be a closed linear subspace of $H$.
Let $P_K$ denote the orthogonal projection on $K$.
Then $P_K$ is bounded.
That is:
- $\norm {\map {P_K} h} \le \norm h$
for each $h \in H$.
Fixed Points of Orthogonal Projection
Let $\struct {H, \innerprod \cdot \cdot}$ be a Hilbert space.
Let $\norm \cdot$ be the inner product norm of $H$.
Let $K$ be a closed linear subspace of $H$.
Let $P_K$ denote the orthogonal projection on $K$.
Let $h \in H$.
Then:
- $\map {P_K} h = h$
if and only if $h \in K$.
Orthogonal Projection is Projection
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.:
- $P_K \circ P_K = P_K$
Kernel of Orthogonal Projection
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:
- $\ker P_K = K^\bot$
where:
- $\ker P_K$ denotes the kernel of $P_K$
- $K^\bot$ denotes the orthocomplement of $K$.
Range of Orthogonal Projection
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:
- $P_K \sqbrk H = K$
where $P_K \sqbrk H$ denotes the image of $H$ under $P_K$.
Orthogonal Projection onto Orthocomplement
Let $H$ be a Hilbert space.
Let $A$ be a closed linear subspace of $H$.
Let $P_A: H \to H$ be the orthogonal projection onto $A$.
Let $P_{A^\perp}: H \to H$ be the orthogonal projection onto $A^\perp$, the orthocomplement of $A$.
Let $I: H \to H$ be the identity operator on $H$.
Then:
- $P_{A^\perp} = I - P_A$
Sources
- 1990: John B. Conway: A Course in Functional Analysis (2nd ed.) ... (previous) ... (next): $\text{I}$ Hilbert Spaces: $\S 2.$ Orthogonality: Theorem $2.7$