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A reflection through an axis. In mathematics, a reflection (also spelled reflexion) [1] is a mapping from a Euclidean space to itself that is an isometry with a hyperplane as the set of fixed points; this set is called the axis (in dimension 2) or plane (in dimension 3) of reflection.
The unit sphere can be replaced with the closed unit ball in the definition. Namely, a normed vector space is uniformly convex if and only if for every < there is some > so that, for any two vectors and in the closed unit ball (i.e. ‖ ‖ and ‖ ‖) with ‖ ‖, one has ‖ + ‖ (note that, given , the corresponding value of could be smaller than the one provided by the original weaker ...
An example of a reflexive relation is the relation "is equal to" on the set of real numbers, since every real number is equal to itself. A reflexive relation is said to have the reflexive property or is said to possess reflexivity. Along with symmetry and transitivity, reflexivity is one of three properties defining equivalence relations.
The promised geometric property of reflexive Banach spaces is the following: if is a closed non-empty convex subset of the reflexive space , then for every there exists a such that ‖ ‖ minimizes the distance between and points of .
A hyper-reflexive operator algebra is automatically reflexive. In the case of a reflexive algebra of matrices with nonzero entries specified by a given pattern, the problem of finding the distance constant can be rephrased as a matrix-filling problem: if we fill the entries in the complement of the pattern with arbitrary entries, what choice of ...
On a finite-dimensional linear space, a quadratic form x↦f(x) is always a (finite) linear combination of products x↦g(x) h(x) of two linear functionals g and h.Therefore, assuming that the scalars are complex numbers, every sequence x n satisfying g(x n) → 0 for all linear functionals g, satisfies also f(x n) → 0 for all quadratic forms f.