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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.
In mathematics, a Hilbert–Schmidt integral operator is a type of integral transform.Specifically, given a domain Ω in n-dimensional Euclidean space R n, then the square-integrable function k : Ω × Ω → C belonging to L 2 (Ω×Ω) such that
In mathematics, a locally integrable function (sometimes also called locally summable function) [1] is a function which is integrable (so its integral is finite) on every compact subset of its domain of definition. The importance of such functions lies in the fact that their function space is similar to L p spaces, but its members are not ...
The log-spectral distance (LSD), also referred to as log-spectral distortion or root mean square log-spectral distance, is a distance measure between two spectra. [1] The log-spectral distance between spectra () and ^ is defined as p-norm:
The Schatten 1-norm is the nuclear norm (also known as the trace norm, or the Ky Fan n-norm [1]). The Schatten 2-norm is the Frobenius norm. The Schatten ∞-norm is the spectral norm (also known as the operator norm, or the largest singular value).
In mathematics, the L p spaces are function spaces defined using a natural generalization of the p-norm for finite-dimensional vector spaces.They are sometimes called Lebesgue spaces, named after Henri Lebesgue (Dunford & Schwartz 1958, III.3), although according to the Bourbaki group (Bourbaki 1987) they were first introduced by Frigyes Riesz ().
The Frobenius norm defined by ‖ ‖ = = = | | = = = {,} is self-dual, i.e., its dual norm is ‖ ‖ ′ = ‖ ‖.. The spectral norm, a special case of the induced norm when =, is defined by the maximum singular values of a matrix, that is, ‖ ‖ = (), has the nuclear norm as its dual norm, which is defined by ‖ ‖ ′ = (), for any matrix where () denote the singular values ...
In mathematics, the operator norm measures the "size" of certain linear operators by assigning each a real number called its operator norm. Formally, it is a norm defined on the space of bounded linear operators between two given normed vector spaces .