This article contains the Hahn Banach theorem in the vector space over the field of complex numbers. provide the terminology and notation for this paper. The following propositions are true: (1) For every element z of C holds ||z|| = |z|. (2) For all elements x 1 , y 1 , x 2 , y 2 of R holds (x 1 + y 1 i) · (x 2 + y 2 i) = (x 1 · x 2 − y 1 · y 2) + (x 1 · y 2 + x 2 · y 1)i. (3) For every real n...

although homotopy-based methods, namely homotopy analysis method andhomotopy perturbation method, have largely been used to solve functionalequations, there are still serious questions on the convergence issue of thesemethods. some authors have tried to prove convergence of these methods, butthe researchers in this article indicate that some of those discussions are faulty.here, after criticizi...

Problem 11 is due to P. E. Howard.’ The results of Problems 12 and 13 were announced by Pincus. The independence result in Theorem 7.1 1 was proved by Levy [I9621 for ZFA and transferred to Z F by Pincus [ 1969,1973b1, whose is also the observation in Problem 15.’ Example 7.12 in Section 7.4 is due to Tarski. Conditions (S) and (D) were formulated by Mostowski [1945], who proved the independenc...

let  be a banach algebra. let  be linear mappings on . first we demonstrate a theorem concerning the continuity of double derivations; especially that all of -double derivations are continuous on semi-simple banach algebras, in certain case. afterwards we define a new vocabulary called “-higher double derivation” and present a relation between this subject and derivations and finally give some ...

We provide a formal, simple and intuitive theory of rational decision making including sequential decisions that affect the environment. The theory has a geometric flavor, which makes the arguments easy to visualize and understand. Our theory is for complete decision makers, which means that they have a complete set of preferences. Our main result shows that a complete rational decision maker i...

In this paper we prove a strong Hahn-Banach theorem: separation of disjoint convex sets by linear forms is possible without any further conditions, if the target field R is replaced by a more general real closed extension field. From this we deduce a general Positivstellensatz for ∗-algebras, involving representations over real closed fields. We investigate the class of group algebras in more d...

We give necessary and sufficient conditions in order that a finite sequence (X1, . . . , Xn) of exchangeable random elements in a fairly general space S be extendible to a longer finite or to an infinite exchangeable sequence. This is done by formulating the extendibility problem as the extension problem for certain bounded linear functionals on suitable normed spaces and by using the Hahn-Bana...

The aim of this paper is to prove the Uniform Boundedness Principle and Banach-Steinhaus Theorem for anti linear operators and hence strong linear operators on Banach hypervector spaces. Also we prove the continuity of the product operation in such spaces.