Quasi-uniform and uniform convergence of Riemann and Riemann-type integrable functions with values in a Banach space

In this article, we study quasi-uniform and uniform convergence of nets and sequences of different types of functions defined on a topological space, in particular, on a closed bounded interval of R, with values in a metric space and in some cases in a Banach space. We show that boundedness and continuity are inherited to the quasi-uniform limit, and integrability is inherited to the uniform limit of a net of functions. Given a sequence of functions, we construct functions with values in a sequence space and consequently we infer some important properties of such functions. Finally, we study convergence of partially equi-regulated* nets of functions which is shown to be a generalized notion of exhaustiveness.

Download Full-text

Injective absolutely summing operators

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100065105 ◽

1988 ◽

Vol 103 (3) ◽

pp. 497-502

Author(s):

Susumu Okada ◽

Yoshiaki Okazaki

Keyword(s):

Banach Space ◽

Uniform Convergence ◽

Convex Space ◽

Locally Convex Space ◽

Dimensional Banach Space ◽

Infinite Dimensional ◽

Infinite Dimensional Banach Space ◽

Integrable Functions ◽

Locally Convex ◽

Convergence In Mean

Let X be an infinite-dimensional Banach space. It is well-known that the space of X-valued, Pettis integrable functions is not always complete with respect to the topology of convergence in mean, that is, the uniform convergence of indefinite integrals (see [14]). The Archimedes integral introduced in [9] does not suffer from this defect. For the Archimedes integral, functions to be integrated are allowed to take values in a locally convex space Y larger than the space X while X accommodates the values of indefinite integrals. Moreover, there exists a locally convex space Y, into which X is continuously embedded, such that the space ℒ(μX, Y) of Y-valued, Archimedes integrable functions is identical to the completion of the space of X valued, simple functions with repect to the toplogy of convergence in mean, for each non-negative measure μ (see [9]).

Download Full-text

Best simultaneous approximation on metric spaces via monotonous norms

Filomat ◽

10.2298/fil2011777k ◽

2020 ◽

Vol 34 (11) ◽

pp. 3777-3787

Author(s):

Mona Khandaqji ◽

Aliaa Burqan

Keyword(s):

Banach Space ◽

Continuous Function ◽

Finite Number ◽

Metric Space ◽

Measure Space ◽

Simultaneous Approximation ◽

Metric Spaces ◽

Closed Subspace ◽

Integrable Functions ◽

Best Simultaneous Approximation

For a Banach space X, L?(T,X) denotes the metric space of all X-valued ?-integrable functions f : T ? X, where the measure space (T,?,?) is a complete positive ?-finite and ? is an increasing subadditive continuous function on [0,?) with ?(0) = 0. In this paper we discuss the proximinality problem for the monotonous norm on best simultaneous approximation from the closed subspace Y?X to a finite number of elements in X.

Download Full-text

A continuity property related to an index of non-separability and its applications

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700011680 ◽

1992 ◽

Vol 46 (1) ◽

pp. 67-79 ◽

Cited By ~ 4

Author(s):

Warren B. Moors

Keyword(s):

Banach Space ◽

Topological Space ◽

Convex Function ◽

Compact Subset ◽

Metric Space ◽

Dense Subset ◽

Countable Family ◽

Set Valued Mapping ◽

Continuous Convex Function ◽

Fréchet Differentiable

For a set E in a metric space X the index of non-separability is β(E) = inf{r > 0: E is covered by a countable-family of balls of radius less than r}.Now, for a set-valued mapping Φ from a topological space A into subsets of a metric space X we say that Φ is β upper semi-continuous at t ∈ A if given ε > 0 there exists a neighbourhood U of t such that β(Φ(U)) < ε. In this paper we show that if the subdifferential mapping of a continuous convex function Φ is β upper semi-continuous on a dense subset of its domain then Φ is Fréchet differentiable on a dense Gδ subset of its domain. We also show that a Banach space is Asplund if and only if every weak* compact subset has weak* slices whose index of non-separability is arbitrarily small.

Download Full-text

On sequences of integrable functions

Journal of the Australian Mathematical Society ◽

10.1017/s1446788700026896 ◽

1962 ◽

Vol 2 (3) ◽

pp. 295-300

Author(s):

Basil C. Rennie

Keyword(s):

Banach Space ◽

Strong Convergence ◽

Finite Measure ◽

Complex Banach Space ◽

The Real ◽

Integrable Functions ◽

Necessary And Sufficient ◽

Sequence Of Functions

Let f1(x), f2(x), … be a sequence of functions belonging to the real or complex Banach space L, (see S. Banach: [1] (The results can be generalised to functions on any space that is the union of countably many sets of finite measure). We are concerned with various properties that such a sequence may have, and in particular with the more important kinds of convergence (strong, weak and pointwise). This article shows what relations connect the various properties considered; for instance that for strong convergence (i.e. ║fn — f║ → 0) it is necessary and sufficient firstly that the sequence should converge weakly (i.e. if g is bounded and measurable then f(fn(x) — f(x))g(x)dx → 0) and secondly that any sub-sequence should contain a sub-sub-sequence converging p.p. to f(x).

Download Full-text

Kadec–Klee properties of vector-valued Hardy spaces

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100075617 ◽

1992 ◽

Vol 111 (3) ◽

pp. 535-544 ◽

Cited By ~ 3

Author(s):

P. N. Dowling ◽

C. J. Lennard

Keyword(s):

Banach Space ◽

Uniform Convergence ◽

Hardy Spaces ◽

Unit Disc ◽

Open Unit ◽

Uniformly Convex ◽

Open Unit Disc ◽

Vector Valued ◽

Sequence Of Functions ◽

Compact Subsets

In 1930, S. Warschawski [19] showed that H1(D), where D is the open unit disc in ℂ, has the following property: Let be a sequence of functions in H1(D) converging uniformly on compact subsets of D to a function f∈H1(D) and suppose that ‖fn‖1 = |f‖1 = 1 for all n∈ℕ. Then converges to zero. From a Banach space standpoint, this result says that H1(D) has the Kadec–Klee property with respect to uniform convergence on compact subsets of D. Warschawski's result was proved independently by Newman [16] in 1963 (see also [13] for another proof) and extended to more general domains by Hoffman [12], Goldstein and Swaminathan [8] and Godefroy [7]. A uniform version of Warschawski's result and its subsequent extensions was recently obtained by Besbes, Dilworth, Dowling and Lennard [2] (see also [1]). We mention here that these results for H1 spaces also hold for the Hp-spaces for 1 < p < ∞ because these spaces are uniformly convex.

Download Full-text

On Uniform Approximations of Abstract Functions

Canadian Mathematical Bulletin ◽

10.4153/cmb-1967-011-9 ◽

1967 ◽

Vol 10 (1) ◽

pp. 99-108

Author(s):

Elias Zakon

Keyword(s):

Metric Space ◽

Real Function ◽

Present Note ◽

Uniform Limit ◽

Uniform Approximations ◽

Range Of Values ◽

Separable Metric Space ◽

Sequence Of Functions

As is well known, every real function is the pointwise (uniform) limit of a sequence of functions with a finite (countable) range of values. Monna [5] and Kvačko [4] suggested some extensions of this theorem to functions with values in a separable metric space. In the present note we give some further generalizations, with an emphasis on uniform approximations which have many applications in the generalized theory of measure and integration. In particular, we consider measurable abstract functions (mappings).

Download Full-text

On the domain of Riesz mean in the space Ls

Filomat ◽

10.2298/fil1704925y ◽

2017 ◽

Vol 31 (4) ◽

pp. 925-940 ◽

Cited By ~ 12

Author(s):

Medine Yeşilkayagil ◽

Feyzi Başar

Keyword(s):

Banach Space ◽

Sequence Space ◽

Double Sequence ◽

Sufficient Conditions ◽

Necessary And Sufficient Conditions ◽

Double Sequences ◽

Riesz Mean ◽

Double Sequence Space ◽

Necessary And Sufficient ◽

Barrelled Space

Let 0 < s < ?. In this study, we introduce the double sequence space Rqt(Ls) as the domain of four dimensional Riesz mean Rqt in the space Ls of absolutely s-summable double sequences. Furthermore, we show that Rqt(Ls) is a Banach space and a barrelled space for 1 ? s < 1 and is not a barrelled space for 0 < s < 1. We determine the ?- and ?(?)-duals of the space Ls for 0 < s ? 1 and ?(bp)-dual of the space Rqt(Ls) for 1 < s < 1, where ? ? {p, bp, r}. Finally, we characterize the classes (Ls:Mu), (Ls:Cbp), (Rqt(Ls) : Mu) and (Rqt(Ls):Cbp) of four dimensional matrices in the cases both 0 < s < 1 and 1 ? s < 1 together with corollaries some of them give the necessary and sufficient conditions on a four dimensional matrix in order to transform a Riesz double sequence space into another Riesz double sequence space.

Download Full-text

On Simultaneous Farthest Points in

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/2011/890598 ◽

2011 ◽

Vol 2011 ◽

pp. 1-10

Author(s):

Sh. Al-Sharif ◽

M. Rawashdeh

Keyword(s):

Banach Space ◽

Bounded Subset ◽

Farthest Points ◽

Integrable Functions

Let be a Banach space and let be a closed bounded subset of . For , we set . The set is called simultaneously remotal if, for any , there exists such that . In this paper, we show that if is separable simultaneously remotal in , then the set of -Bochner integrable functions, , is simultaneously remotal in . Some other results are presented.

Download Full-text

Characterizations and properties of graphs of Baire functions

Mathematica Slovaca ◽

10.1515/ms-2017-0145 ◽

2018 ◽

Vol 68 (4) ◽

pp. 789-802

Author(s):

Balázs Maga

Keyword(s):

Banach Space ◽

Topological Space ◽

Convex Set ◽

Vertical Section ◽

Fréchet Space ◽

Frechet Space ◽

On Domain of Nörlund Sequences

Mathematics ◽

10.3390/math6110268 ◽

2018 ◽

Vol 6 (11) ◽

pp. 268 ◽

Cited By ~ 2

Author(s):

Kuddusi Kayaduman ◽

Fevzi Yaşar

Keyword(s):

Banach Space ◽

Bounded Variation ◽

Sequence Space ◽

Convergent Series ◽

Sequence Spaces ◽

Matrix Transformations ◽

Nörlund Matrix ◽

Inclusion Relations ◽

The Matrix ◽

New Space

In 1978, the domain of the Nörlund matrix on the classical sequence spaces lp and l∞ was introduced by Wang, where 1 ≤ p < ∞. Tuğ and Başar studied the matrix domain of Nörlund mean on the sequence spaces f0 and f in 2016. Additionally, Tuğ defined and investigated a new sequence space as the domain of the Nörlund matrix on the space of bounded variation sequences in 2017. In this article, we defined new space and and examined the domain of the Nörlund mean on the bs and cs, which are bounded and convergent series, respectively. We also examined their inclusion relations. We defined the norms over them and investigated whether these new spaces provide conditions of Banach space. Finally, we determined their α, β, γduals, and characterized their matrix transformations on this space and into this space.

Download Full-text