Search results
Results from the WOW.Com Content Network
If a vector space is not finite-dimensional, then its (algebraic) dual space is always of larger dimension (as a cardinal number) than the original vector space. This is in contrast to the case of the continuous dual space, discussed below, which may be isomorphic to the original vector space even if the latter is infinite-dimensional.
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 ...
3.2 Finite-dimensional spaces. ... the dual norm is a measure of size for a continuous linear function defined on a normed ... this dual space is also a Hilbert ...
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, = ¯.
For any measurable space, the finite measures form a convex cone in the Banach space of signed measures with the total variation norm. Important subsets of the finite measures are the sub-probability measures, which form a convex subset, and the probability measures, which are the intersection of the unit sphere in the normed space of signed measures and the finite measures.
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 ...
This space is the strong dual space of ... on an underlying measure space, (, ... by a finite vector, and the consumption set is a vector space with a finite ...
A measure space is a basic object of measure theory, a branch of mathematics that studies generalized notions of volumes. It contains an underlying set, the subsets of this set that are feasible for measuring (the σ-algebra) and the method that is used for measuring (the measure). One important example of a measure space is a probability space.