The orthogonal complement of a subspace V of the vector space R^n is the set of vectors which are orthogonal to all elements of V. For example, the orthogonal complement of the space generated by two non proportional vectors u, v of the real space R^3 is the subspace formed by all normal vectors to the plane spanned by u and v. In general, any subspace V of an inner product space E has an orthogonal complement V^⊥ and E = V⊕V^⊥. This property extends to any subspace V of a space E equipped with a symmetric or differential k-form or a Hermitian form which is nonsingular on V.

Fredholm's theorem | orthogonal decomposition | orthogonal sum