Difference between revisions of "Orthogonal complement"

Latest revision as of 04:07, 8 April 2017

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

1. ↑
   TODO: Can we relax this to a subset maybe?
2. ↑ 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]

References

1. ↑ Warwick 2014 Lecture Notes - Functional Analysis - Richard Sharp