Properties of Orthogonal Projection

Theorem

Orthogonal Projection is Linear Transformation

Let $\GF \in \set {\R, \C}$.

This article, or a section of it, needs explaining. In particular: Does this hold for all subfields of $\C$? You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Explain}} from the code.

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 $\HH$ 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.

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 $2.7$