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For a finite-dimensional inner product space of dimension , the orthogonal complement of a -dimensional subspace is an ()-dimensional subspace, and the double orthogonal complement is the original subspace: =.
In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space [1] [2]) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar , often denoted with angle brackets such as in a , b {\displaystyle \langle a,b\rangle } .
Given an inner product space V, we can form the orthogonal complement F(X ) of any subspace X of V. This yields an antitone Galois connection between the set of subspaces of V and itself, ordered by inclusion; both polarities are equal to F .
Two vector subspaces and of an inner product space are called orthogonal subspaces if each vector in is orthogonal to each vector in . The largest subspace of V {\displaystyle V} that is orthogonal to a given subspace is its orthogonal complement .
A real inner product space is defined in the same way, except that H is a real vector space and the inner product takes real values. Such an inner product will be a bilinear map and (,, , ) will form a dual system. [5]
When the vector space has an inner product and is complete (is a Hilbert space) the concept of orthogonality can be used. An orthogonal projection is a projection for which the range U {\displaystyle U} and the kernel V {\displaystyle V} are orthogonal subspaces .
When V is an inner product space, the quotient / can be identified with the orthogonal complement in V of (). This is the generalization to linear operators of the row space , or coimage, of a matrix.
For a general inner product space , an orthonormal basis can be used to define normalized orthogonal coordinates on . Under these coordinates, the inner product becomes a dot product of vectors. Thus the presence of an orthonormal basis reduces the study of a finite-dimensional inner product space to the study of R n {\displaystyle \mathbb {R ...