The fact that every column vector in is orthogonal to every column vector in can be checked by direct computation. The fact that the spans of these vectors are orthogonal then follows by bilinearity of the dot product. Finally, the fact that these spaces are orthogonal complements follows from the dimension relationships given below.
General bilinear formsEdit
Let be a vector space over a field equipped with a bilinear form We define to be left-orthogonal to , and to be right-orthogonal to when For a subset of define the left orthogonal complement to be
If are subspaces of a finite-dimensional space and then
Inner product spacesEdit
This section considers orthogonal complements in an inner product space
Two vectors and are called orthogonal if which happens if and only if for all scalars 
If is any subset of an inner product space then its orthogonal complement in is the vector subspace
which is always a closed subset of [proof 1] that satisfies and if then also and
If is a vector subspace of an inner product space then
If is a closed vector subspace of a Hilbert space then
where is called the orthogonal decomposition of into and and it indicates that is a complemented subspace of with complement
The orthogonal complement is always closed in the metric topology. In finite-dimensional spaces, that is merely an instance of the fact that all subspaces of a vector space are closed. In infinite-dimensional Hilbert spaces, some subspaces are not closed, but all orthogonal complements are closed. If is a vector subspace of an inner product space the orthogonal complement of the orthogonal complement of is the closure of that is,
Some other useful properties that always hold are the following. Let be a Hilbert space and let and be its linear subspaces. Then:
if then ;
if is a closed linear subspace of then ;
if is a closed linear subspace of then the (inner) direct sum.
For a finite-dimensional inner product space of dimension the orthogonal complement of a -dimensional subspace is an -dimensional subspace, and the double orthogonal complement is the original subspace:
There is a natural analog of this notion in general Banach spaces. In this case one defines the orthogonal complement of W to be a subspace of the dual of V defined similarly as the annihilator
It is always a closed subspace of V∗. There is also an analog of the double complement property. W⊥⊥ is now a subspace of V∗∗ (which is not identical to V). However, the reflexive spaces have a naturalisomorphismi between V and V∗∗. In this case we have
^If then which is closed in so assume Let where is the underlying scalar field of and define by which is continuous because this is true of each of its coordinates Then is closed in because is closed in and is continuous. If is linear in its first (respectively, its second) coordinate then is a linear map (resp. an antilinear map); either way, its kernel is a vector subspace of Q.E.D.