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Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. 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.