Could someone explain to me, using perhaps a very simple example in @d, what we mean by orthogonal projection from space D to space D'? Thanks
I am not sure about your notation but a simple example goes as follows. Consider $D=\mathbb R^2$, i.e. the 2-dimensional plane and as $D'$ some line through the origin lying in this plane. Now the projection from $D$ to $D'$ is just a map which assignes a point of $D'$ to each point of $D$ as follows:
Take any point in $D$, drop a perpendicular to $D'$, the point where it hits $D'$ is your image.
Note that any point which is originally in $D'$ stays unchanged.