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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 ...
A Buekenhout geometry consists of a set X whose elements are called "varieties", with a symmetric, reflexive relation on X called "incidence", together with a function τ called the "type map" from X to a set Δ whose elements are called "types" and whose size is called the "rank". Two distinct varieties of the same type cannot be incident.
A topological vector space that is canonically isomorphic to its bidual ″ is called a reflexive space: ″. Examples: As in the finite-dimensional case, on each Hilbert space H its inner product ⋅, ⋅ defines a map H → H ∗ , v ↦ ( w ↦ w , v ) , {\displaystyle H\to H^{*},v\mapsto (w\mapsto \langle w,v\rangle ),} which is a bijection ...
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.
In a non-reflexive Banach space, such as the Lebesgue space () of all bounded sequences, Riesz’s lemma does not hold for =. [ 5 ] However, every finite dimensional normed space is a reflexive Banach space, so Riesz’s lemma does holds for α = 1 {\displaystyle \alpha =1} when the normed space is finite-dimensional, as will now be shown.
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).
The reflector is the completion of a metric space on objects, and the extension by density on arrows. [1]: 90 The category of sheaves is a reflective subcategory of presheaves on a topological space. The reflector is sheafification, which assigns to a presheaf the sheaf of sections of the bundle of its germs.