and Applied Analysis 3 is Cauchy. If each Cauchy sequence is convergent, then the fuzzy norm is said to be complete, and the fuzzy normed space is called a fuzzy Banach space. Let X,N be a fuzzy normed space and Y,N ′ a fuzzy Banach space. For a given mapping f : X → Y , we use the abbreviation Df ( x, y ) : f ( 2x y ) f ( 2x − y 2f x − fx y − fx − y − 2f 2x , 2.1 for all x, y ∈ X. Recall Df ≡ ...

This paper investigates the fractals generated by the dynamical system of intuitionistic fuzzy contractions in the intuitionistic fuzzy metric spaces by generalizing the Hutchinson-Barnsley theory. We prove some existence and uniqueness theorems of fractals in the standard intuitionistic fuzzy metric spaces by using the intuitionistic fuzzy Banach contraction theorem. In addition to that, we an...

z on the classical Hardy space H. Though there are many aspects of this operator worthy of study [20], we will focus on the description of its invariant subspaces by which we mean the closed linear manifolds E ⊂ H for which BE ⊂ E . When 1 < p < ∞, a seminal paper of Douglas, Shapiro, and Shields [8] describes these invariant subspaces by using the important concept of a pseudocontinuation deve...

We present a complete formalization of the Hahn-Banach theorem in the simply-typed set-theory of Isabelle/HOL, such that both the modeling of the underlying mathematical notions and the full proofs are intelligible to human readers. This is achieved by means of the Isar environment, which provides a framework for high-level reasoning based on natural deduction. The final result is presented as ...

In this paper we study the existence and characterization of spaces which are images of minimal-norm projections that are required to interpolate at given functionals and satisfy additional shape-preserving requirements. We will call such spaces optimal interpolating spaces preserving shape. This investigation leads to concrete solutions in classical settings and, as examples, n will be determi...

In contrast to construction of integrals as limits of Riemann sums, the Gelfand-Pettis characterization is a property no reasonable notion of integral would lack. Since this property is an irreducible minimum, this definition of integral is called a weak integral. Uniqueness of the integral is immediate when the dual V ∗ separates points, meaning that for v 6 v′ in V there is λ ∈ V ∗ with λv 6=...

We show in this paper that a certain class of normed modules over the algebra of all bounded operators on a Hilbert space possesses a homological property which is a kind of a functional-analytic version of the standard algebraic property of flatness. We mean the preservation, under projective tensor multiplication of modules, of the property of a given morphism to be isometric. As an applicati...

The question whether or not the sum of two maximal monotone operators is maximal monotone under Rockafellar’s constraint qualification— that is, whether or not “the sum theorem” is true— is themost famous open problem inMonotone Operator Theory. In his 2008monograph “From Hahn-Banach to Monotonicity”, Stephen Simons asked whether or not the sum theorem holds for the special case of a maximal mo...

We discuss three major classes of theorems in additive and extremal com-binatorics: decomposition theorems, approximate structure theorems, and transference principles. We also show how the finite-dimensional Hahn-Banach theorem can be used to give short and transparent proofs of many results of these kinds. Amongst the applications of this method is a much shorter proof of one of the major ste...