Orthogonal complement
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Contents
Definition
Let [ilmath]((X,[/ilmath][ilmath]\mathbb{K} [/ilmath][ilmath]),\langle\cdot,\cdot\rangle)[/ilmath] be an inner-product space and let [ilmath]L[/ilmath] be a vector subspace of the vector space [ilmath](X,\mathbb{K})[/ilmath][Note 1], then we may define the orthogonal complement of [ilmath]L[/ilmath], denoted [ilmath]L^\perp[/ilmath], as follows[1]:
- [ilmath]L^\perp:\eq\Big\{x\in X\ \Big\vert\ \forall y\in L[[/ilmath][ilmath]\langle x,y\rangle\eq 0[/ilmath][ilmath]]\ \Big\} [/ilmath] - notice that [ilmath]\langle x,y\rangle\eq 0[/ilmath] is the definition of [ilmath]x[/ilmath] and [ilmath]y[/ilmath] being orthogonal vectors, thus:
- the orthogonal complement is all vectors which are orthogonal to the entire of [ilmath]L[/ilmath].
Properties
- The orthogonal complement of a vector subspace is a vector subspace - [ilmath]L^\perp[/ilmath] is a vector subspace of [ilmath](X,\mathbb{K})[/ilmath].
- The orthogonal complement of a vector subspace is a topologically closed set - [ilmath]L^\perp[/ilmath] is a closed set with respect to the topology induced by the inner product[Note 2]
Notes
- ↑ TODO: Can we relax this to a subset maybe?
- ↑ The topology we consider [ilmath](X,\langle\cdot,\cdot\rangle)[/ilmath] with is the topology induced by the metric [ilmath]d(x,y):\eq\Vert x-y\Vert[/ilmath] which is the metric induced by the norm [ilmath]\Vert x\Vert:\eq \sqrt{\langle x,x\rangle} [/ilmath] which itself is the norm induced by the inner product [ilmath]\langle\cdot,\cdot\rangle[/ilmath]
