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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 ...
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 Cotlar–Stein almost orthogonality lemma is a mathematical lemma in the field of functional analysis. It may be used to obtain information on the operator norm on an operator , acting from one Hilbert space into another, when the operator can be decomposed into almost orthogonal pieces.
In Euclidean space, two vectors are orthogonal if and only if their dot product is zero, i.e. they make an angle of 90° (radians), or one of the vectors is zero. [4] Hence orthogonality of vectors is an extension of the concept of perpendicular vectors to spaces of any dimension.
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 other words, the Stiefel manifold () is the orthogonal, unitary, or symplectic frame bundle associated to the tautological bundle on a Grassmannian. When one passes to the n → ∞ {\displaystyle n\to \infty } limit, these bundles become the universal bundles for the classical groups.
Eidelheit theorem: A Fréchet space is either isomorphic to a Banach space, or has a quotient space isomorphic to . Kadec renorming theorem: Every separable Banach space X {\displaystyle X} admits a Kadec norm with respect to a countable total subset A ⊆ X ∗ {\displaystyle A\subseteq X^{*}} of X ∗ . {\displaystyle X^{*}.}
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.