Orthogonal Projections in Hilbert Spaces

Question

Let $H$ be a Hilbert Space and let $K \subset H$ be a non-empty closed convex set in $H.$ Let $P_K: H \mapsto K$ be the orthogonal projection of $H$ onto $K,$ so $P_K(u)$ is the distance minimizing vector in $K,$ namely,

$$||P_K(u)-u||_H\leq ||v-u||_H$$

for all $v \in K.$

I have two questions:

1. In general, is this map linear ( the text I am reading makes no mention of this and I am wondering if it is well known or not true)? If not, is there a topological or linear-algebraic classification of spaces on which this map is linear. (I am aware that this map is non-expansive and continuous in the norm on $H$).

2. Is there an orthogonality (with respect to the inner product on $H$) relationship between $P_K(x)$ and $P_K(-x).$

This is not homework, it just pertains to something I am currently studying. Thank you for the help.