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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.
If and are two finite-dimensional normed spaces with the same dimension, let (,) denote the collection of all linear isomorphisms :. Denote by ‖ ‖ the operator norm of such a linear map — the maximum factor by which it "lengthens" vectors.
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.
Write I(S) for the set of integral linear combinations of S, and I 0 (S) for the subset of degree 0 elements of I(S). Suppose that τ is an isometry from I 0 (S) to the degree 0 virtual characters of G. Then τ is called coherent if it can be extended to an isometry from I(S) to characters of G and I 0 (S) is non-zero.
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.
Having no linear functionals is highly undesirable for the purposes of doing analysis. In case of the Lebesgue measure on R n , {\displaystyle \mathbb {R} ^{n},} rather than work with L p {\displaystyle L^{p}} for 0 < p < 1 , {\displaystyle 0<p<1,} it is common to work with the Hardy space H p whenever possible, as this has quite a few linear ...
An isometry V is said to be pure if, in the notation of the above proof, = {}. The multiplicity of a pure isometry V is the dimension of the kernel of V*, i.e. the cardinality of the index set A in the Wold decomposition of V. In other words, a pure isometry of multiplicity N takes the form
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 ...