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This is a list of functional analysis topics. See also: Glossary of functional analysis. Hilbert space ... Normed vector space; Unit ball; Banach space; Hahn–Banach ...
It was simply referred to as property (H) in a list of properties for normed spaces that starts with (A) and ends with (H). This list was given by K. Fan and I. Glicksberg (Observe that the definition of (H) given by Fan and Glicksberg includes additionally the rotundity of the norm, so it does not coincide with the Radon-Riesz property itself).
As the standard negator is used in the above definition of a t-norm/t-conorm pair, this can be generalized as follows: A De Morgan triplet is a triple (T,⊥,n) such that [1] T is a t-norm; ⊥ is a t-conorm according to the axiomatic definition of t-conorms as mentioned above; n is a strong negator
However, every finite dimensional normed space is a reflexive Banach space, so Riesz’s lemma does holds for = when the normed space is finite-dimensional, as will now be shown. When the dimension of X {\displaystyle X} is finite then the closed unit ball B ⊆ X {\displaystyle B\subseteq X} is compact.
In functional analysis and related areas of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topological vector spaces whose topology is generated by translations of balanced, absorbent, convex sets.
Banach lattices are extremely common in functional analysis, and "every known example [in 1948] of a Banach space [was] also a vector lattice." [1] In particular: ℝ, together with its absolute value as a norm, is a Banach lattice.
In the mathematical study of functional analysis, the Banach–Mazur distance is a way to define a distance on the set () of -dimensional normed spaces. With this distance, the set of isometry classes of n {\displaystyle n} -dimensional normed spaces becomes a compact metric space , called the Banach–Mazur compactum .
The other method is used if the disk is absorbing: in this case, the auxiliary normed space is the quotient space / (). If the disk is both bounded and absorbing then the two auxiliary normed spaces are canonically isomorphic (as topological vector spaces and as normed spaces).