What exactly is a non-linear orthogonal projection?

Question

In a Hilbert space of bounded integrable functions, let $P$ be an operator such that

$$P(f(x)) = \frac{f(x)+|f(x)|}{2}$$

The complement of $P$ can be written as $Q = I - P$, hence

$$Q(f(x)) = \frac{f(x)-|f(x)|}{2}$$

Both $P$ and $Q$ are idempotent. However, in contrary to many textbook examples, $P$ and $Q$ are not linear, nor are they Hermitian. Still

$$\langle P(f(x))|Q(f(x))\rangle = 0$$

holds. Are we allowed to call $P$ an orthogonal projection and $Q$ its orthogonal complement?

(Sorry for my inaccurate wording, I am a physicist, not a mathematician.)

Answer 1

After a lot of Googling, I am able to come up with an answer myself. There is a well-known definition of non-linear orthogonal projections. Roughly, the procedure is as follows:

• In a metric space (e.g. a Hilbert space $H$), choose any non-empty subset $M$, which typically might be a manifold, but need not necessarily be a linear subspace.
• For each element $h \in H$, define a so-called distance function $$\rho(h,M)=\inf_{m \in M} ||h-m||$$.
• In the domain of all $h$ that have a unique $m$ (i.e. a perpendicular foot point in $M$), we can define a projection $P$ such that $P(h) = m$.

To point it out explicitly, the properties of $P$ totally depend on the norm $||\cdot||$ and on the choice of $M$. As a simple example, let $M$ be the unit circle in the complex plain. Then $P(z)=z/|z|$.

If, as a special case, $M$ is a closed linear subspace of $H$, then $P$ turns out to be a linear orthogonal projection.

For more details, see e.g. http://matwbn.icm.edu.pl/ksiazki/apm/apm59/apm5911.pdf.

Answer 2

A linear projection maps each point in a space onto the nearest point (according to some norm) of a linear subspace.

A non-linear projection maps each point in a space onto the nearest point (according to some norm) of a non-linear subspace (e.g. some manifold).

To give an example, a linear projection may map each point in three-dimensional space onto a plane whereas a non-linear projection may map each point in three-dimensional space onto a sphere or any other manifold.