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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 ...
The concept of unit circle (the set of all vectors of norm 1) is different in different norms: for the 1-norm, the unit circle is a square oriented as a diamond; for the 2-norm (Euclidean norm), it is the well-known unit circle; while for the infinity norm, it is an axis-aligned square.
For example, the L 2-norm gives rise to the Cramér–von Mises statistic. The asymptotic distribution can be further characterized in several different ways. First, the central limit theorem states that pointwise , F ^ n ( t ) {\displaystyle \scriptstyle {\widehat {F}}_{n}(t)} has asymptotically normal distribution with the standard n ...
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The Euclidean norm of a Euclidean vector space is a special case that allows defining Euclidean distance by the formula (,) = ‖ ‖. The study of normed spaces and Banach spaces is a fundamental part of functional analysis , a major subfield of mathematics.
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A probability distribution is not uniquely determined by the moments E[X n] = e nμ + 1 / 2 n 2 σ 2 for n ≥ 1. That is, there exist other distributions with the same set of moments. [4] In fact, there is a whole family of distributions with the same moments as the log-normal distribution. [citation needed]
For example, points (2, 0), (2, 1), and (2, 2) lie along the perimeter of a square and belong to the set of vectors whose sup norm is 2. In mathematical analysis, the uniform norm (or sup norm) assigns, to real-or complex-valued bounded functions defined on a set , the non-negative number