Orthogonal complement
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In the mathematical fields of linear algebra and functional analysis, the orthogonal complement 
of a subspace W of an inner product space V is the set of all vectors in V that are orthogonal to every vector in W, i.e., it is
The orthogonal complement is always closed in the metric topology. In Hilbert spaces, the orthogonal complement of the orthogonal complement of W is the closure of W, i.e.,
[edit] Banach spaces
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 by
It is always a closed subspace of V * . There is also an analog of the double complement property. 
is now a subspace of 
(which is not identical to V). However, the reflexive spaces have a natural isomorphism i between V and 
. In this case we have
This is a rather straightforward consequence of the Hahn-Banach theorem.
