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However, every finite dimensional normed space is a reflexive Banach space, so Riesz’s lemma does holds for = when the normed space is finite-dimensional, as will now be shown. When the dimension of X {\displaystyle X} is finite then the closed unit ball B ⊆ X {\displaystyle B\subseteq X} is compact.
The Gram-Schmidt theorem, together with the axiom of choice, guarantees that every vector space admits an orthonormal basis. This is possibly the most significant use of orthonormality, as this fact permits operators on inner-product spaces to be discussed in terms of their action on the space's orthonormal basis vectors. What results is a deep ...
Let stand for ,, or . The Stiefel manifold () can be thought of as a set of n × k matrices by writing a k-frame as a matrix of k column vectors in . The orthonormality condition is expressed by A*A = where A* denotes the conjugate transpose of A and denotes the k × k identity matrix.
Depending on the bilinear form, the vector space may contain null vectors, non-zero self-orthogonal vectors, in which case perpendicularity is replaced with hyperbolic orthogonality. In the case of function spaces , families of functions are used to form an orthogonal basis , such as in the contexts of orthogonal polynomials , orthogonal ...
The Riesz–Fischer theorem also applies in a more general setting. Let R be an inner product space consisting of functions (for example, measurable functions on the line, analytic functions in the unit disc; in old literature, sometimes called Euclidean Space), and let {} be an orthonormal system in R (e.g. Fourier basis, Hermite or Laguerre polynomials, etc. – see orthogonal polynomials ...
In finite-dimensional spaces, the matrix representation (with respect to an orthonormal basis) of an orthogonal transformation is an orthogonal matrix. Its rows are mutually orthogonal vectors with unit norm, so that the rows constitute an orthonormal basis of V. The columns of the matrix form another orthonormal basis of V.
The norm on induced by , is equal to the original norm on and the continuous dual space of is the set of all real-valued bounded -linear functionals on (see the article about the polarization identity for additional details about this relationship).
The Hahn–Banach theorem is a central tool in functional analysis.It allows the extension of bounded linear functionals defined on a vector subspace of some vector space to the whole space, and it also shows that there are "enough" continuous linear functionals defined on every normed vector space to make the study of the dual space "interesting".