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Encyclopedia > Orthogonal complement

In the mathematical fields of linear algebra and functional analysis, the orthogonal complement 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 W, i.e. it is

In infinite-dimensional Hilbert spaces, it is of some interest to observe that every orthogonal complement is closed in the metric topology—a statement that is vacuously true in the finite-dimensional case. The orthogonal complement of the orthogonal complement of W is the closure of W, i.e.,

Banach spaces

There is a natural analog of this notion in general Banach spaces. In this case one defines the orthogonal complemet of W to be a subspace of the dual of V defined similarly by

It is always a closed subspace of V * . There is also an analog of the double complement property. is now a subspace of which is not identical to V. However, the reflexive spaces have a natural isomorphism i between V and . In this case we have

This is a rather straightforward consequence of the Hahn-Banach theorem.


  Results from FactBites:
 
Orthogonality - Wikipedia, the free encyclopedia (1348 words)
Hence orthogonality of vectors is a generalization of the concept of perpendicular.
In 4D the orthogonal complement of a line is a hyperplane and vice versa, and that of a plane is a plane.
The Legendre polynomials are orthogonal with respect to the uniform distribution on the interval from −1 to 1.
  More results at FactBites »


 

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