scholarly journals New Inner Product Quasilinear Spaces on Interval Numbers

2016 ◽  
Vol 2016 ◽  
pp. 1-9 ◽  
Author(s):  
Hacer Bozkurt ◽  
Yılmaz Yilmaz
Keyword(s):  
Functional Analysis ◽  
Inner Product ◽  
Interval Numbers ◽  
Algebraic Properties ◽  
Interval Space

Primarily we examine the new example of quasilinear spaces, namely, “IRninterval space.” We obtain some new theorems and results related to this new quasilinear space. After giving some new notions of quasilinear dependence-independence and basis on quasilinear functional analysis, we obtain some results onIRninterval space related to these concepts. Secondly, we presentIs,Ic0,Il∞, andIl2quasilinear spaces and we research some algebraic properties of these spaces. We obtain some new results and provide an important contribution to the improvement of quasilinear functional analysis.

scholarly journals Some Logical Metatheorems with Applications in Functional Analysis

2003 ◽  
Vol 10 (21) ◽  
Author(s):  
Ulrich Kohlenbach
Keyword(s):  
Functional Analysis ◽  
Fixed Point ◽  
Metric Spaces ◽  
Fixed Point Theory ◽  
Point Theory ◽  
Inner Product ◽  
Original Proof ◽  
Uniform Bounds ◽  
Existence Proofs ◽  
Special Cases

In previous papers we have developed proof-theoretic techniques for extracting effective uniform bounds from large classes of ineffective existence proofs in functional analysis. `Uniform' here means independence from parameters in compact spaces. A recent case study in fixed point theory systematically yielded uniformity even w.r.t. parameters in metrically bounded (but noncompact) subsets which had been known before only in special cases. In the present paper we prove general logical metatheorems which cover these applications to fixed point theory as special cases but are not restricted to this area at all. Our theorems guarantee under general logical conditions such strong uniform versions of non-uniform existence statements. Moreover, they provide algorithms for actually extracting effective uniform bounds and transforming the original proof into one for the stronger uniformity result. Our metatheorems deal with general classes of spaces like metric spaces, hyperbolic spaces, normed linear spaces, uniformly convex spaces as well as inner product spaces.

scholarly journals Some Inequalities in Functional Analysis, Combinatorics, and Probability Theory

10.37236/330 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Chunrong Feng ◽  
Liangpan Li ◽  
Jian Shen
Keyword(s):  
Probability Theory ◽  
Functional Analysis ◽  
Finite Field ◽  
Best Approximation ◽  
Approximation Theorem ◽  
Inner Product ◽  
Product Spaces ◽  
Inner Product Spaces ◽  
The Best Approximation

The main purpose of this paper is to show that many inequalities in functional analysis, probability theory and combinatorics are immediate corollaries of the best approximation theorem in inner product spaces. Besides, as applications of the de Caen-Selberg inequality, the finite field Kakeya and Nikodym problems are also studied.

scholarly journals General Logical Metatheorems for Functional Analysis

2005 ◽  
Vol 12 (21) ◽  
Author(s):  
Philipp Gerhardy ◽  
Ulrich Kohlenbach
Keyword(s):  
Functional Analysis ◽  
Fixed Point Theory ◽  
Point Theory ◽  
Inner Product ◽  
Product Spaces ◽  
Nonlinear Functional Analysis ◽  
Nonlinear Functional ◽  
Large Classes ◽  
Global Boundedness ◽  
Uniformly Continuous

In this paper we prove general logical metatheorems which state that for large classes of theorems and proofs in (nonlinear) functional analysis it is possible to extract from the proofs effective bounds which depend only on very sparse local bounds on certain parameters. This means that the bounds are uniform for all parameters meeting these weak local boundedness conditions. The results vastly generalize related theorems due to the second author where the global boundedness of the underlying metric space (resp. a convex subset of a normed space) was assumed. Our results treat general classes of spaces such as metric, hyperbolic, CAT(0), normed, uniformly convex and inner product spaces and classes of functions such as nonexpansive, Hölder-Lipschitz, uniformly continuous, bounded and weakly quasi-nonexpansive ones. We give several applications in the area of metric fixed point theory. In particular, we show that the uniformities observed in a number of recently found effective bounds (by proof theoretic analysis) can be seen as instances of our general logical results.

On Maps Preserving Products

2005 ◽  
Vol 48 (3) ◽  
pp. 355-369 ◽  
Author(s):  
M. A. Chebotar ◽  
W.-F. Ke ◽  
P.-H. Lee ◽  
L.-S. Shiao
Keyword(s):  
Functional Analysis ◽  
Linear Algebra ◽  
Point Of View ◽  
Algebraic Properties ◽  
Theoretic Point

AbstractMaps preserving certain algebraic properties of elements are often studied in Functional Analysis and Linear Algebra. The goal of this paper is to discuss the relationships among these problems from the ring-theoretic point of view.

scholarly journals Quasilinear Inner Product Spaces and Hilbert Quasilinear Spaces

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Hacer Bozkurt ◽  
Sümeyye Çakan ◽  
Yılmaz Yılmaz
Keyword(s):  
Functional Analysis ◽  
Nonlinear Analysis ◽  
Linear Functional ◽  
Fréchet Derivative ◽  
Inner Product ◽  
Product Spaces ◽  
Frechet Derivative ◽  
Inner Product Spaces

Aseev launched a new branch of functional analysis by introducing the theory of quasilinear spaces in the framework of the topics of norm, bounded quasilinear operators and functionals (Aseev (1986)). Furthermore, some quasilinear counterparts of classical nonlinear analysis that lead to such result as Frechet derivative and its applications were examined deal with. This pioneering work causes a lot of results in such applications such as (Rojas-Medar et al. (2005), Talo and Başar (2010), and Nikol'skiĭ (1993)). His work has motivated us to introduce the concept of quasilinear inner product spaces. Thanks to this new notion, we obtain some new theorems and definitions which are quasilinear counterparts of fundamental definitions and theorems in linear functional analysis. We claim that some new results related to this concept provide an important contribution to the improvement of quasilinear functional analysis.

scholarly journals A note on near-rings of mappings

1973 ◽  
Vol 16 (2) ◽  
pp. 214-215 ◽  
Author(s):  
Sadayuki Yamamuro
Keyword(s):  
Functional Analysis ◽  
Algebraic Theory ◽  
Algebraic Properties ◽  
Non Linear

Hanna Neumann was one of the pioneers of the algebraic theory of nearrings. In the functional analysis, the near-rings appear as the sets of non-linear mappings of a space into itself. The relations between the algebraic properties of these near-rings and the analytic structures of the spaces and mappings involved have not been fully investigated. The purpose of this note is to consider one of such problems.

Case Formulation and Design of Behavioral Treatment Programs

2003 ◽  
Vol 19 (3) ◽  
pp. 164-174 ◽  
Author(s):  
Stephen N. Haynes ◽  
Andrew E. Williams
Keyword(s):  
Functional Analysis ◽  
Behavior Problems ◽  
Behavioral Treatment ◽  
Clinical Case ◽  
Relative Magnitude ◽  
Mechanisms Of Action ◽  
Treatment Programs ◽  
Case Formulation ◽  
Functional Relations ◽  
Causal Variables

Summary: We review the rationale for behavioral clinical case formulations and emphasize the role of the functional analysis in the design of individualized treatments. Standardized treatments may not be optimally effective for clients who have multiple behavior problems. These problems can affect each other in complex ways and each behavior problem can be influenced by multiple, interacting causal variables. The mechanisms of action of standardized treatments may not always address the most important causal variables for a client's behavior problems. The functional analysis integrates judgments about the client's behavior problems, important causal variables, and functional relations among variables. The functional analysis aids treatment decisions by helping the clinician estimate the relative magnitude of effect of each causal variable on the client's behavior problems, so that the most effective treatments can be selected. The parameters of, and issues associated with, a functional analysis and Functional Analytic Clinical Case Models (FACCM) are illustrated with a clinical case. The task of selecting the best treatment for a client is complicated because treatments differ in their level of specificity and have unequally weighted mechanisms of action. Further, a treatment's mechanism of action is often unknown.

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