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Suppose a vector norm ‖ ‖ on and a vector norm ‖ ‖ on are given. Any matrix A induces a linear operator from to with respect to the standard basis, and one defines the corresponding induced norm or operator norm or subordinate norm on the space of all matrices as follows: ‖ ‖, = {‖ ‖: ‖ ‖ =} = {‖ ‖ ‖ ‖:} . where denotes the supremum.
In mathematics, the logarithmic norm is a real-valued functional on operators, and is derived from either an inner product, a vector norm, or its induced operator norm. The logarithmic norm was independently introduced by Germund Dahlquist [ 1 ] and Sergei Lozinskiĭ in 1958, for square matrices .
In mathematics, a norm is a function from a real or complex vector space to the non-negative real numbers that behaves in certain ways like the distance from the origin: it commutes with scaling, obeys a form of the triangle inequality, and is zero only at the origin.
Every real -by-matrix corresponds to a linear map from to . Each pair of the plethora of (vector) norms applicable to real vector spaces induces an operator norm for all -by-matrices of real numbers; these induced norms form a subset of matrix norms.
The space of distributions, being defined as the continuous dual space of (), is then endowed with the (non-metrizable) strong dual topology induced by () and the canonical LF-topology (this topology is a generalization of the usual operator norm induced topology that is placed on the continuous dual spaces of normed spaces).
The norm induced by this inner product is the Hilbert–Schmidt norm under which the space of Hilbert–Schmidt operators is complete (thus making it into a Hilbert space). [4] The space of all bounded linear operators of finite rank (i.e. that have a finite-dimensional range) is a dense subset of the space of Hilbert–Schmidt operators (with ...
Here φ∈L 2 (G/H) means: the space G/H carries a suitable invariant measure, and since the norm of φ(g) is constant on each left coset of H, we can integrate the square of these norms over G/H and obtain a finite result. The group G acts on the induced representation space by translation, that is, (g.φ)(x)=φ(g −1 x) for g,x∈G and φ∈ ...
Banach space, normed vector spaces which are complete with respect to the metric induced by the norm; Banach–Mazur compactum – Concept in functional analysis; Finsler manifold, where the length of each tangent vector is determined by a norm; Inner product space, normed vector spaces where the norm is given by an inner product