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Banach Spaces
Published in J. Tinsley Oden, Leszek F. Demkowicz, Applied Functional Analysis, 2017
J. Tinsley Oden, Leszek F. Demkowicz
Before we proceed with some general results concerning dual spaces, we present in this section a few nontrivial examples of dual spaces in the form of so-called representation theorems. The task of a representation theorem is to identify elements from a dual space (i.e., linear and continuous functionals defined on a normed space) with elements from some other space, for instance some other functions, through a representation formula relating functionals with those functions. The representation theorems not only provide meaningful characterizations of dual spaces, but are also of great practical value in applications.
On pressure-driven Hele–Shaw flow of power-law fluids
Published in Applicable Analysis, 2022
John Fabricius, Salvador Manjate, Peter Wall
Since B is a continuous isomorphism, we can identify where is the annihilator of defined as and by the Riesz Representation theorem, we have . The dual operator of B is usually called De Rham's operator or the ‘pressure operator’. Note that is also continuous with . To emphasize that B depends on the domain , we write . Thus, to obtain a -bound for the pressure we need to investigate how the operator norm depends on ε.
Two notions of MV-algebraic semisimplicity relative to fixed MV-chains
Published in Journal of Applied Non-Classical Logics, 2022
Celestin Lele, Jean B. Nganou, Jean M. Wagoum
Suppose that A is a Boolean algebra. Then by Birkhoff's representation theorem for Boolean algebras (Birkhoff, 1935, Thm. 21), A is a sub-MV-algebra of a powerset MV-algebra for some nonempty set X, where denotes the 2-element Boolean algebra. For each , consider be the natural projection. Then, for each x. Clearly, , which implies that and A is -semisimple.
Fifty years of similarity relations: a survey of foundations and applications
Published in International Journal of General Systems, 2022
One of the most interesting issues related to fuzzy equivalence relations is the way they can be generated, which depends on the way in which the data are given and on the use we want to make of them. It also helps to understand their structure. The four most common ways are: By calculating the T-transitive closure of a reflexive and symmetric fuzzy relation (a proximity or tolerance relation).By using the Representation Theorem.By generating a decomposable operator from a fuzzy subset.By obtaining a transitive opening of a proximity relation.