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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.
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.
If V is a vector space with a quadratic form Q, then the conformal orthogonal group CO(V, Q) is the group of linear transformations T of V for which there exists a scalar λ such that for all x in V Q ( T x ) = λ 2 Q ( x ) {\displaystyle Q(Tx)=\lambda ^{2}Q(x)}
In mathematics, the Mazur–Ulam theorem states that if and are normed spaces over R and the mapping: is a surjective isometry, then is affine.It was proved by Stanisław Mazur and Stanisław Ulam in response to a question raised by Stefan Banach.
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 ...
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.
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