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A Banach space is super-reflexive if all Banach spaces finitely representable in are reflexive, or, in other words, if no non-reflexive space is finitely representable in . The notion of ultraproduct of a family of Banach spaces [ 14 ] allows for a concise definition: the Banach space X {\displaystyle X} is super-reflexive when its ultrapowers ...
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 ...
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.
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.
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 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).
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.
In universal algebra and lattice theory, a tolerance relation on an algebraic structure is a reflexive symmetric relation that is compatible with all operations of the structure. Thus a tolerance is like a congruence , except that the assumption of transitivity is dropped. [ 1 ]