We have seen several sequences and series whose terms depended on a parameter or variable.Such se... more We have seen several sequences and series whose terms depended on a parameter or variable.Such sequences are \(a^n\), \(\root n \of {a}\), \(a^n /n!\), \(\left( 1+\frac{a}{n} \right) ^n\) and every power series. Now we turn to the systematic investigation of those sequences and series that depend on a variable.
A definitive positive answer to the so-called proper subspace problem (a.k.a. the atomic space pr... more A definitive positive answer to the so-called proper subspace problem (a.k.a. the atomic space problem) for quasi-normed spaces is given: every infinite dimensional quasi-normed space has a proper closed infinite-dimensional subspace.
A sequence (f n ) is called a sequence of functions if for each n ∈ N, its n-th term is a real-va... more A sequence (f n ) is called a sequence of functions if for each n ∈ N, its n-th term is a real-valued function.
In this paper, we prove a strong convergence of Ishikawa scheme to a uniformly L-Lipschitzian and... more In this paper, we prove a strong convergence of Ishikawa scheme to a uniformly L-Lipschitzian and asymptotically pseudocontractive mappings in the intermediate sense. No compactness assumption is imposed either on T or C , and computation of intersection of closed convex sets C n and Q n for each n ≥ 1 is not required. We also obtain convergence results in this direction for asymptotically strict pseudocontractive mappings in the intermediate sense. Our theorems improve and unify most of the results that have been proved for this important class of nonlinear mappings.
International Journal of Applied Physics and Mathematics, 2019
We investigate the common features and the resemblance of the central parts of the different exis... more We investigate the common features and the resemblance of the central parts of the different existing integration theories to obtain a more unified approach to the notion of integral. Our approach gives a presentation of the integral that does not require the development of measure theory.
British Journal of Mathematics & Computer Science, 2014
We introduce the notion of summable bases that naturally generalizes the notion of unconditional ... more We introduce the notion of summable bases that naturally generalizes the notion of unconditional sequence bases for Banach spaces. We shall be particularly interested in some classical results on sequences and series in separable Banach spaces that carry over or naturally extend to the case of non-separable Banach spaces.
Abstract We present a Riesz representation theorem in the setting of extended integration theory ... more Abstract We present a Riesz representation theorem in the setting of extended integration theory as introduced in [6]. The result is used to obtain boundedness theorems for integral operators in the more general setting of spaces of vector valued extended integrable functions.
British Journal of Mathematics & Computer Science, 2014
We use tensor product to introduce a new approach to the theory of integration. Such an approach ... more We use tensor product to introduce a new approach to the theory of integration. Such an approach will strengthen the existing various classical concepts of integral and will provide a continuous thread tying the subject matter together. The integral of vector-valued functions with respect to vector-valued additive measures will be covered without any assumption of measurability. As applications, we state and prove extensions of the Lebesgue fundamental theorems of convergence in a more general setting.
British Journal of Mathematics & Computer Science, 2015
We give a non-metric version of the Besicovitch Covering Lemma and an extension of the Lebesgue-D... more We give a non-metric version of the Besicovitch Covering Lemma and an extension of the Lebesgue-Differentiation Theorem into the setting of the integration theory of vector valued functions.
In previous chapters, we considered the algebraic and order structures ℝ of the set of real numbe... more In previous chapters, we considered the algebraic and order structures ℝ of the set of real numbers. We have seen that properties such as convergence of sequences, continuity and differentiability of functions, are all defined locally around some given point. In this chapter, we shall study ℝ further by considering its features locally around each of its points. The generalization of such an approach to a more abstract setting is of great importance in analysis. In this chapter, we will discuss only subsets of the real line.
Although the Riemann integral finds many useful applications, we will see in this chapter that it... more Although the Riemann integral finds many useful applications, we will see in this chapter that it has some great limitations. The class of Riemann integrable functions is “incomplete” in some sense. We will build upon these limitations of the Riemann integral to introduce a more far-reaching theory of integration: the Lebesgue integral.1
British Journal of Mathematics & Computer Science, 2015
We revisit the definition of the tensor integrability introduced in [1], and prove some useful ch... more We revisit the definition of the tensor integrability introduced in [1], and prove some useful characterizations that provide a foundation for studying relevant properties to the integration theory. As applications, we obtain some representation theorems for the projective and injective tensor products of the space of scalar integrable functions with Banach spaces.
The Stone-Weierstrass theorem is concerned about approximation of continuous functions by polynom... more The Stone-Weierstrass theorem is concerned about approximation of continuous functions by polynomials. In this chapter,1 we discuss yet another type of approximation which applies to functions that are not necessarily continuous. The applications of the kind of approximation we are going to study here are of considerable importance especially in physics and engineering.
Integration theory has always been at the center of any analysis-related courses. This explains t... more Integration theory has always been at the center of any analysis-related courses. This explains the many different approaches to this theory. In this chapter, we discuss the somewhat traditional approach of the Riemann integral and some other closely related topics.
Numbers and functions are both basic and fundamental concepts in mathematics. In this chapter, we... more Numbers and functions are both basic and fundamental concepts in mathematics. In this chapter, we will study the system of real numbers, emphasizing the essential properties from the axioms of a complete ordered field. We also recall the rudiments of the notion of functions without going into the details.
We have seen several sequences and series whose terms depended on a parameter or variable.Such se... more We have seen several sequences and series whose terms depended on a parameter or variable.Such sequences are \(a^n\), \(\root n \of {a}\), \(a^n /n!\), \(\left( 1+\frac{a}{n} \right) ^n\) and every power series. Now we turn to the systematic investigation of those sequences and series that depend on a variable.
A definitive positive answer to the so-called proper subspace problem (a.k.a. the atomic space pr... more A definitive positive answer to the so-called proper subspace problem (a.k.a. the atomic space problem) for quasi-normed spaces is given: every infinite dimensional quasi-normed space has a proper closed infinite-dimensional subspace.
A sequence (f n ) is called a sequence of functions if for each n ∈ N, its n-th term is a real-va... more A sequence (f n ) is called a sequence of functions if for each n ∈ N, its n-th term is a real-valued function.
In this paper, we prove a strong convergence of Ishikawa scheme to a uniformly L-Lipschitzian and... more In this paper, we prove a strong convergence of Ishikawa scheme to a uniformly L-Lipschitzian and asymptotically pseudocontractive mappings in the intermediate sense. No compactness assumption is imposed either on T or C , and computation of intersection of closed convex sets C n and Q n for each n ≥ 1 is not required. We also obtain convergence results in this direction for asymptotically strict pseudocontractive mappings in the intermediate sense. Our theorems improve and unify most of the results that have been proved for this important class of nonlinear mappings.
International Journal of Applied Physics and Mathematics, 2019
We investigate the common features and the resemblance of the central parts of the different exis... more We investigate the common features and the resemblance of the central parts of the different existing integration theories to obtain a more unified approach to the notion of integral. Our approach gives a presentation of the integral that does not require the development of measure theory.
British Journal of Mathematics & Computer Science, 2014
We introduce the notion of summable bases that naturally generalizes the notion of unconditional ... more We introduce the notion of summable bases that naturally generalizes the notion of unconditional sequence bases for Banach spaces. We shall be particularly interested in some classical results on sequences and series in separable Banach spaces that carry over or naturally extend to the case of non-separable Banach spaces.
Abstract We present a Riesz representation theorem in the setting of extended integration theory ... more Abstract We present a Riesz representation theorem in the setting of extended integration theory as introduced in [6]. The result is used to obtain boundedness theorems for integral operators in the more general setting of spaces of vector valued extended integrable functions.
British Journal of Mathematics & Computer Science, 2014
We use tensor product to introduce a new approach to the theory of integration. Such an approach ... more We use tensor product to introduce a new approach to the theory of integration. Such an approach will strengthen the existing various classical concepts of integral and will provide a continuous thread tying the subject matter together. The integral of vector-valued functions with respect to vector-valued additive measures will be covered without any assumption of measurability. As applications, we state and prove extensions of the Lebesgue fundamental theorems of convergence in a more general setting.
British Journal of Mathematics & Computer Science, 2015
We give a non-metric version of the Besicovitch Covering Lemma and an extension of the Lebesgue-D... more We give a non-metric version of the Besicovitch Covering Lemma and an extension of the Lebesgue-Differentiation Theorem into the setting of the integration theory of vector valued functions.
In previous chapters, we considered the algebraic and order structures ℝ of the set of real numbe... more In previous chapters, we considered the algebraic and order structures ℝ of the set of real numbers. We have seen that properties such as convergence of sequences, continuity and differentiability of functions, are all defined locally around some given point. In this chapter, we shall study ℝ further by considering its features locally around each of its points. The generalization of such an approach to a more abstract setting is of great importance in analysis. In this chapter, we will discuss only subsets of the real line.
Although the Riemann integral finds many useful applications, we will see in this chapter that it... more Although the Riemann integral finds many useful applications, we will see in this chapter that it has some great limitations. The class of Riemann integrable functions is “incomplete” in some sense. We will build upon these limitations of the Riemann integral to introduce a more far-reaching theory of integration: the Lebesgue integral.1
British Journal of Mathematics & Computer Science, 2015
We revisit the definition of the tensor integrability introduced in [1], and prove some useful ch... more We revisit the definition of the tensor integrability introduced in [1], and prove some useful characterizations that provide a foundation for studying relevant properties to the integration theory. As applications, we obtain some representation theorems for the projective and injective tensor products of the space of scalar integrable functions with Banach spaces.
The Stone-Weierstrass theorem is concerned about approximation of continuous functions by polynom... more The Stone-Weierstrass theorem is concerned about approximation of continuous functions by polynomials. In this chapter,1 we discuss yet another type of approximation which applies to functions that are not necessarily continuous. The applications of the kind of approximation we are going to study here are of considerable importance especially in physics and engineering.
Integration theory has always been at the center of any analysis-related courses. This explains t... more Integration theory has always been at the center of any analysis-related courses. This explains the many different approaches to this theory. In this chapter, we discuss the somewhat traditional approach of the Riemann integral and some other closely related topics.
Numbers and functions are both basic and fundamental concepts in mathematics. In this chapter, we... more Numbers and functions are both basic and fundamental concepts in mathematics. In this chapter, we will study the system of real numbers, emphasizing the essential properties from the axioms of a complete ordered field. We also recall the rudiments of the notion of functions without going into the details.
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