Generalizations
More generally, given a map between normed vector spaces one can analogously ask for this map to be an isometry on the orthogonal complement of the kernel: that be an isometry; in particular it must be onto. The case of an orthogonal projection is when W is a subspace of V. In Riemannian geometry, this is used in the definition of a Riemannian submersion.
Read more about this topic: Projection (linear Algebra)