On the Existence Theorems of Kantorovich, Moore and Miranda

~ An extension of S mulian's theorem in which neither the linearity nor the local convexity are presupposed is obtained. As a consequence, a non-Archimedean version of S mulian's theorem is derived. Definition [ 1] Let Jf..( be a metrizable topological division ring, G a vector space o ver Ji -<:, Ha set and fa mapping from G xH into K such taht for each hE H, the mapping f¡,: gEG Hjh (g) =f(g,h) E K is linear, and such that f separates points of G (that is, for all gE G, g:;t:O, there is hE H with fh (g) :;t: 0). The initial topology on g for the family 7H = lfh : h E

2009

Let T be a compact Hausdor topological space and let M denote an n-dimensional subspace of the space C(T ), the space of real-valued continuous functions on T and let the space be equipped with the uniform norm. Zukhovitskii (7) attributes the Basic Theorem to E.Ya.Remez and gives a proof by duality. He also gives a proof due to Shnirel&#39;man, which uses Helly&#39;s Theorem, now the paper obtains a new proof of the Basic Theorem. The significance of the Basic Theorem for us is that it reduces the characterization of a best approximation to f 2 C(T ) from M to the case of finite T , that is to the case of approximation in l 1 (r). If one solves the problem for the finite case of T then one can deduce the solution to the general case. An immediate consequence of the Basic Theorem is that for a finite dimensional subspace M of C0(T ) there exists a separating measure for M and f 2 C0(T )\M, the cardinality of whose support is not greater than dimM+1. This result is a special case of ...

2007

We show that many recent and new results on the existence of ACL homeomorphic solutions for the Beltrami equation follow from our extension of the well‐known Lehto existence theorem.

Theor Decis, 1983

Theory and Decision 15 (1983) 309-320. 0040-5833/83/0154-0309501.80. 9 1983 by 1). ReidelPublishing Company.

Communications on Pure and Applied Mathematics, 2003

2002

2003

We prove the following extension of the Mean Value Theorem. Let E be a Banach space and let F : [a, b] → E and ϕ : [a, b] → R be two functions for which there exists a subset A ⊂ [a, b] such that: i) F and ϕ have negligible variation on A,

Review of Symbolic Logic, Vol. 2 (3), 509-516.

Three-valued accounts of conditionals frequently promise (a) to conform to the probabilistic view that conditionals are evaluated by conditional probabilities, and (b) to yield a plausible account of compounds of conditionals. However, McGee (1981) shows that probabilistic validity, the conception of validity most naturally associated with the probabilistic view, cannot be characterized by a finite matrix. Adams (1995) indicates a further generalization of this result. Nevertheless, Adams (1986) provides a description of probabilistic validity in three-valued terms by going beyond the standard framework. Yet the language Adams considers is severely restricted: it does not contain compounds of conditionals. Thus, a natural question arises: Is there a plausible three-valued account of compounds of conditionals which agrees with probabilistic validity on the restricted language? In this note, I develop a general framework in which to address this question. The answer will be negative.

Proceedings of the American Mathematical Society, 1981

A new direct proof of a theorem of Stanojević is given. Consequently it is proven that the Fomin class F p ( 1 &gt; p ⩽ 2 ) {\mathcal {F}_p}(1 &gt; p \leqslant 2) is a subclass of the class B V ∩ C \mathcal {B}\mathcal {V} \cap \mathcal {C} , where C \mathcal {C} is the Garrett-Stanojević class and B V \mathcal {B}\mathcal {V} is the class of null sequences of bounded variation. This also provides a new direct proof of Fomin’s theorem.

Tokyo Journal of Mathematics, 1984