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A Banach space is reflexive if it is linearly isometric to its bidual under this canonical embedding . James' space is an example of a non-reflexive space which is linearly isometric to its bidual. Furthermore, the image of James' space under the canonical embedding J {\displaystyle J} has codimension one in its bidual.
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 ...
This operation is also known as a central inversion (Coxeter 1969, §7.2), and exhibits Euclidean space as a symmetric space. In a Euclidean vector space, the reflection in the point situated at the origin is the same as vector negation. Other examples include reflections in a line in three-dimensional space.
A reflexive, weak, [1] or non-strict partial order, [2] commonly referred to simply as a partial order, is a homogeneous relation ≤ on a set that is reflexive, antisymmetric, and transitive. That is, for all a , b , c ∈ P , {\displaystyle a,b,c\in P,} it must satisfy:
The equipollence relation between line segments in geometry is a common example of an equivalence relation. A simpler example is equality. A simpler example is equality. Any number a {\displaystyle a} is equal to itself (reflexive).
In mathematics, more specifically in functional analysis, a Banach space (/ ˈ b ɑː. n ʌ x /, Polish pronunciation:) is a complete normed vector space.Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vectors and is complete in the sense that a Cauchy sequence of vectors always converges to a well-defined limit that is ...
A locally convex space is called semi-reflexive if the evaluation map : (′) ′ is surjective (hence bijective); it is called reflexive if the evaluation map : (′) ′ is surjective and continuous, in which case J will be an isomorphism of TVSs).
An example of a left quasi-reflexive relation is a left Euclidean relation, which is always left quasi-reflexive but not necessarily right quasi-reflexive, and thus not necessarily quasi-reflexive. An example of a coreflexive relation is the relation on integers in which each odd number is related to itself and there are no other relations. The ...