Orthogonal complement

Stub grade: B

This page is a stub

This page is a stub, so it contains little or minimal information and is on a to-do list for being expanded.The message provided is:

Proper stub Alec (talk) 04:07, 8 April 2017 (UTC)

Definition

Let $((X,$$\mathbb{K}$$),\langle\cdot,\cdot\rangle)$ be an inner-product space and let $L$ be a vector subspace of the vector space $(X,\mathbb{K})$, then we may define the orthogonal complement of $L$, denoted $L^\perp$, as follows:

• $L^\perp:\eq\Big\{x\in X\ \Big\vert\ \forall y\in L[$$\langle x,y\rangle\eq 0$$]\ \Big\}$ - notice that $\langle x,y\rangle\eq 0$ is the definition of $x$ and $y$ being orthogonal vectors, thus:
  • the orthogonal complement is all vectors which are orthogonal to the entire of $L$.

Properties

• The orthogonal complement of a vector subspace is a vector subspace - $L^\perp$ is a vector subspace of $(X,\mathbb{K})$.
• The orthogonal complement of a vector subspace is a topologically closed set - $L^\perp$ is a closed set with respect to the topology induced by the inner product