Concept

Operator norm

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Singular value decomposition

In linear algebra, the singular value decomposition (SVD) is a factorization of a real or complex matrix. It generalizes the eigendecomposition of a square normal matrix with an orthonormal eigenbasis to any matrix. It is related to the polar decomposition. Specifically, the singular value decomposition of an complex matrix M is a factorization of the form where U is an complex unitary matrix, is an rectangular diagonal matrix with non-negative real numbers on the diagonal, V is an complex unitary matrix, and is the conjugate transpose of V.

Continuous linear operator

In functional analysis and related areas of mathematics, a continuous linear operator or continuous linear mapping is a continuous linear transformation between topological vector spaces. An operator between two normed spaces is a bounded linear operator if and only if it is a continuous linear operator. Continuous function (topology) and Discontinuous linear map Bounded operator Suppose that is a linear operator between two topological vector spaces (TVSs). The following are equivalent: is continuous.

Dual norm

In functional analysis, the dual norm is a measure of size for a continuous linear function defined on a normed vector space. Let be a normed vector space with norm and let denote its continuous dual space. The dual norm of a continuous linear functional belonging to is the non-negative real number defined by any of the following equivalent formulas: where and denote the supremum and infimum, respectively.

Bounded set (topological vector space)

In functional analysis and related areas of mathematics, a set in a topological vector space is called bounded or von Neumann bounded, if every neighborhood of the zero vector can be inflated to include the set. A set that is not bounded is called unbounded. Bounded sets are a natural way to define locally convex polar topologies on the vector spaces in a dual pair, as the polar set of a bounded set is an absolutely convex and absorbing set. The concept was first introduced by John von Neumann and Andrey Kolmogorov in 1935.

Spectral radius

In mathematics, the spectral radius of a square matrix is the maximum of the absolute values of its eigenvalues. More generally, the spectral radius of a bounded linear operator is the supremum of the absolute values of the elements of its spectrum. The spectral radius is often denoted by ρ(·). Let λ1, ..., λn be the eigenvalues of a matrix A ∈ Cn×n. The spectral radius of A is defined as The spectral radius can be thought of as an infimum of all norms of a matrix.

Bounded operator

In functional analysis and operator theory, a bounded linear operator is a linear transformation between topological vector spaces (TVSs) and that maps bounded subsets of to bounded subsets of If and are normed vector spaces (a special type of TVS), then is bounded if and only if there exists some such that for all The smallest such is called the operator norm of and denoted by A bounded operator between normed spaces is continuous and vice versa. The concept of a bounded linear operator has been extended from normed spaces to all topological vector spaces.

James's theorem

In mathematics, particularly functional analysis, James' theorem, named for Robert C. James, states that a Banach space is reflexive if and only if every continuous linear functional's norm on attains its supremum on the closed unit ball in A stronger version of the theorem states that a weakly closed subset of a Banach space is weakly compact if and only if the dual norm each continuous linear functional on attains a maximum on The hypothesis of completeness in the theorem cannot be dropped.

Square root of a matrix

In mathematics, the square root of a matrix extends the notion of square root from numbers to matrices. A matrix B is said to be a square root of A if the matrix product BB is equal to A. Some authors use the name square root or the notation A1/2 only for the specific case when A is positive semidefinite, to denote the unique matrix B that is positive semidefinite and such that BB = BTB = A (for real-valued matrices, where BT is the transpose of B).

Singular value

In mathematics, in particular functional analysis, the singular values, or s-numbers of a compact operator acting between Hilbert spaces and , are the square roots of the (necessarily non-negative) eigenvalues of the self-adjoint operator (where denotes the adjoint of ). The singular values are non-negative real numbers, usually listed in decreasing order (σ1(T), σ2(T), ...). The largest singular value σ1(T) is equal to the operator norm of T (see Min-max theorem).

Multiplication operator

In operator theory, a multiplication operator is an operator Tf defined on some vector space of functions and whose value at a function φ is given by multiplication by a fixed function f. That is, for all φ in the domain of Tf, and all x in the domain of φ (which is the same as the domain of f). This type of operator is often contrasted with composition operators. Multiplication operators generalize the notion of operator given by a diagonal matrix.