Editing Orthogonal complement (section)

=== Properties ===

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 space]]s, some subspaces are not closed, but all orthogonal complements are closed. If <math>W</math> is a vector subspace of a [[Hilbert space]] the orthogonal complement of the orthogonal complement of <math>W</math> is the [[Closure (topology)|closure]] of <math>W,</math> that is,
<math display="block">\left(W^\bot\right)^\bot = \overline W.</math>

Some other useful properties that always hold are the following. Let <math>H</math> be a Hilbert space and let <math>X</math> and <math>Y</math> be linear subspaces. Then:
* <math>X^\bot = \overline{X}^{\bot}</math>;
* if <math>Y \subseteq X</math> then <math>X^\bot \subseteq Y^\bot</math>;
* <math>X \cap X^\bot = \{ 0 \}</math>;
* <math>X \subseteq (X^\bot)^\bot</math>;
* if <math>X</math> is a closed linear subspace of <math>H</math> then <math>(X^\bot)^\bot = X</math>;
* if <math>X</math> is a closed linear subspace of <math>H</math> then <math>H = X \oplus X^\bot,</math> the (inner) [[direct sum]].

The orthogonal complement generalizes to the [[Annihilator (ring theory)|annihilator]], and gives a [[Galois connection]] on subsets of the inner product space, with associated [[closure operator]] the topological closure of the span.