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The dual space as defined above is defined for all vector spaces, and to avoid ambiguity may also be called the algebraic dual space. When defined for a topological vector space, there is a subspace of the dual space, corresponding to continuous linear functionals, called the continuous dual space.
Specifically, this dual space is the space of Radon measures on (regular Borel measures), denoted by (). This space, with the norm given by the total variation of a measure, is also a Banach space belonging to the class of ba spaces. (Dunford & Schwartz 1958, §IV.6.3)
i.e. the space of finitely additive signed measures on Σ that are absolutely continuous with respect to μ (μ-a.c. for short). When the measure space is furthermore sigma-finite then L ∞ (μ) is in turn dual to L 1 (μ), which by the Radon–Nikodym theorem is identified with the set of all countably additive μ-a.c. measures. In other ...
Denote by (resp. by ¯) the set of all continuous linear (resp. continuous antilinear) functionals on , which is called the (continuous) dual space (resp. the (continuous) anti-dual space) of . [1] If = then linear functionals on are the same as antilinear functionals and consequently, the same is true for such continuous maps: that is, = ¯.
To distinguish the ordinary dual space from the continuous dual space, the former is sometimes called the algebraic dual space. In finite dimensions, every linear functional is continuous, so the continuous dual is the same as the algebraic dual, but in infinite dimensions the continuous dual is a proper subspace of the algebraic dual.
What follows are two results which will imply that an extended signed measure is the difference of two non-negative measures, and a finite signed measure is the difference of two finite non-negative measures. The Hahn decomposition theorem states that given a signed measure μ, there exist two measurable sets P and N such that: P∪N = X and P ...
There are many closely related variations of the theorem, as the linear functionals can be complex, real, or positive, the space they are defined on may be the unit interval or a compact space or a locally compact space, the continuous functions may be vanishing at infinity or have compact support, and the measures can be Baire measures or ...
In mathematics (specifically in measure theory), a Radon measure, named after Johann Radon, is a measure on the σ-algebra of Borel sets of a Hausdorff topological space X that is finite on all compact sets, outer regular on all Borel sets, and inner regular on open sets. [1]