orthogonal complement
Given a subspace W of an inner product space V, its orthogonal complement W-perp is the set of all vectors perpendicular to everything in W: W-perp = { v in V : <v, w> = 0 for all w in W }. It is the cleanest possible notion of the part of V that points entirely away from W.
W-perp is itself always a subspace (linear combinations of perpendicular vectors stay perpendicular), and W intersect W-perp = {0} since a vector orthogonal to itself has zero norm. In finite dimensions the headline result is the orthogonal decomposition V = W (+) W-perp: every vector splits uniquely as a piece in W plus a piece in W-perp.
That direct-sum splitting is what makes orthogonal projection well-defined and turns geometry into algebra: dim W + dim(W-perp) = dim V, and (W-perp)-perp = W (in finite dimensions). The complement is an honest involution — perp-ing twice returns you home.
A caution for infinite dimensions: the decomposition V = W (+) W-perp needs W to be CLOSED (complete), and (W-perp)-perp gives the closure of W, not W itself. The finite-dimensional case hides these subtleties because every subspace is automatically closed.
The finite-dimensional package: a direct-sum split, dimensions that add up, and a perp operation that undoes itself.
The decomposition V = W (+) W-perp is the algebraic skeleton of dropping a perpendicular in geometry: every vector has a shadow in W and a leftover orthogonal to W, and the two pieces are unique.