High Quality Content by WIKIPEDIA articles! In the mathematical fields of linear algebra and functional analysis, the orthogonal complement W 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 WThe 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.The orthogonal complement generalizes to the annihilator, and gives a Galois connection on subsets of the inner product space, with associated closure operator the topological closure of the span.

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