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In the case that W is the subspace of (with the usual dot product) spanned by the rows of the next matrix,
its orthogonal complement W? is spanned by the three row-vectors of
The fact that every vector on the first list is orthogonal to every vector on the second list 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 forms
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 we define the left orthogonal complement to be
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. In such spaces, the orthogonal complement of the orthogonal complement of is the closure of , i.e.,
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 n, the orthogonal complement of a k-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