I want to know the difference between metric projector and orthogonal projector?

Question

Given a metric space $(X, \rho)$ and $A$ be its closed subset. Now for every $x \in X$ define $$P_A(x) = \{ y \in A : \rho(x, A) = \rho(y, x)\}$$ Now definition of metric projector is as follows: Let $$P_A: x \to P_A(x) : X \to A$$ be the multivalued mapping is metric projecton of $M$ onto $A$ and $P_A$ is orthogonal projector. Now i am not able to find out the basic difference between metric projector and orthogonal projector. Can anyone help me?

Answer 1

I have got some idea about it. Since every Hilbert space $H$ is a metric space with metric induced by the norm. So on a Hilbert space both the projectors on a subspace $M$ are same as if it is closed subspace of $H$ i.e. if $P$ and $P_c$ are othogonal and metric projector on $M$ from $H$, we have $P = P_c$ because unique element exist satisfying given criteria.

Now here comes the difference if $M$ is not a subspace of $H$ but any closed subset of $H$, then we can talk of metric projector only but not of orthogonal projector.