Ads
related to: example of reflexive property in geometry proofs
Search results
Results From The WOW.Com Content Network
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 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 ...
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 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:
In mathematics, the Milman–Pettis theorem states that every uniformly convex Banach space is reflexive. The theorem was proved independently by D. Milman (1938) and B. J. Pettis (1939). S. Kakutani gave a different proof in 1939, and John R. Ringrose published a shorter proof in 1959.
If the normed space X is complete and satisfies the slightly stronger property of being uniformly convex (which implies strict convexity), then it is also reflexive by Milman–Pettis theorem. Properties
In mathematics, a polynomially reflexive space is a Banach space X, on which the space of all polynomials in each degree is a reflexive space. Given a multilinear functional M n of degree n (that is, M n is n -linear), we can define a polynomial p as
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.