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Consequently, the dual space is an important concept in functional analysis. Early terms for dual include polarer Raum [Hahn 1927], espace conjugué, adjoint space [Alaoglu 1940], and transponierter Raum [Schauder 1930] and [Banach 1932]. The term dual is due to Bourbaki 1938. [1]
In mathematics, specifically in order theory and functional analysis, the order dual of an ordered vector space is the set where denotes the set of all positive linear functionals on , where a linear function on is called positive if for all , implies () [1] The order dual of is denoted by +.
Bornologies and Functional Analysis: Introductory Course on the Theory of Duality Topology-Bornology and its use in Functional Analysis. North-Holland Mathematics Studies. Vol. 26. Amsterdam New York New York: North Holland. ISBN 978-0-08-087137-0. MR 0500064. OCLC 316549583. Jarchow, Hans (1981). Locally convex spaces. Stuttgart: B.G. Teubner.
In mathematics, a dual system, dual pair or a duality over a field is a triple (,,) consisting of two vector spaces, and , over and a non-degenerate bilinear map:.. In mathematics, duality is the study of dual systems and is important in functional analysis.
The space of bounded linear operators B(X) on a Banach space X is an example of a unital Banach algebra. Since the definition of the spectrum does not mention any properties of B ( X ) except those that any such algebra has, the notion of a spectrum may be generalised to this context by using the same definition verbatim.
The Erdős–Kaplansky theorem is a theorem from functional analysis.The theorem makes a fundamental statement about the dimension of the dual spaces of infinite-dimensional vector spaces; in particular, it shows that the algebraic dual space is not isomorphic to the vector space itself.
In functional analysis and related areas of mathematics, the strong dual space of a topological vector space (TVS) is the continuous dual space ′ of equipped with the strong (dual) topology or the topology of uniform convergence on bounded subsets of , where this topology is denoted by (′,) or (′,).
Functional analysis is a branch of mathematical analysis, ... contrary to the finite-dimensional situation. This is explained in the dual space article.