We are concerned with establishing completeness and separability criteria for large classes of topological vector spaces which are typically non-locally convex, including Lebesgue-like spaces, Lorentz spaces, Orlicz spaces, mixed-normed spaces, tent spaces, and discrete Triebel Lizorkin and Besov spaces.
Nevertheless, there has been some work on methods of constructing nonstandard models of type theory and models in which various forms of extensionality fail, models for theories with arbitrary (possibly incomplete) sets of logical constants, and on developing general methods of establishing completeness of various systems of axioms with respect to various classes of models.
We establish a completeness theorem.
Similarly to the previous case, we can show that the operator T ˜ f = ∑ n v n ( f ) ( e i λ n t − e i μ n t ), ∀ f ∈ L p t, is bounded in L p t. Introducing the new system { e i ω n t } n ∈ Z in the same manner we establish the completeness of the system (1) in L p t, if λ i ≠ λ j for i ≠ j.
The database was designed to confirm the validity of the identification codes, establish the completeness of the information keyed in and to perform basic data checks.
This study aims to establish the completeness and accuracy of smoking status recording in patients' primary care medical records and the level of interest in receiving smoking cessation support amongst primary care patients in an inner city UK population.
In 2008, Qui and Shu [12] established the completeness of (mathcal{CB}(X)) with respect to the completeness of the metric space X, where (mathcal{CB}(X)) denotes the class of fuzzy sets with nonempty bounded closed α-cut sets equipped with the generalized Hausdorff metric (d_{infty}), which takes the supremum on the Hausdorff distances between the corresponding α-cut sets.
An alternative proof, developed by Skolem in 1922 to avoid appealing to the principles of set theory, has turned out to be useful also for establishing the completeness of the calculus.
