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A path isometry or arcwise isometry is a map which preserves the lengths of curves; such a map is not necessarily an isometry in the distance preserving sense, and it need not necessarily be bijective, or even injective. [5] [6] This term is often abridged to simply isometry, so one should take care to determine from context which type is intended.
It is a linear isometry at the tangent space of every point on (), that is, it is an isometry on the infinitesimal patches. It preserves the curvature tensor at the tangent space of every point on B r ( x ) {\displaystyle B_{r}(x)} , that is, it preserves how the infinitesimal patches fit together.
With this distance, the set of isometry classes of -dimensional normed spaces becomes a compact metric space, called the Banach–Mazur compactum. Definitions [ edit ]
An isometry between two normed vector spaces is a linear map which preserves the norm (meaning ‖ ‖ = ‖ ‖ for all vectors ). Isometries are always continuous and injective . A surjective isometry between the normed vector spaces V {\displaystyle V} and W {\displaystyle W} is called an isometric isomorphism , and V {\displaystyle V} and W ...
In mathematics, a unitary transformation is a linear isomorphism that preserves the inner product: the inner product of two vectors before the transformation is equal to their inner product after the transformation.
In linear algebra, the restricted isometry property (RIP) characterizes matrices which are nearly orthonormal, at least when operating on sparse vectors. The concept was introduced by Emmanuel Candès and Terence Tao [ 1 ] and is used to prove many theorems in the field of compressed sensing . [ 2 ]
In mathematics, particularly in operator theory, Wold decomposition or Wold–von Neumann decomposition, named after Herman Wold and John von Neumann, is a classification theorem for isometric linear operators on a given Hilbert space. It states that every isometry is a direct sum of copies of the unilateral shift and a unitary operator.
A discrete isometry group is an isometry group such that for every point of the space the set of images of the point under the isometries is a discrete set. In pseudo-Euclidean space the metric is replaced with an isotropic quadratic form ; transformations preserving this form are sometimes called "isometries", and the collection of them is ...