Paranormed sequence spaces generated by infinite matrices

1968 ◽  
Vol 64 (2) ◽  
pp. 335-340 ◽  
Author(s):  
I. J. Maddox
Keyword(s):  
Linear Space ◽  
Sequence Spaces ◽  
Topological Linear Space ◽  
Infinite Matrices ◽  
Paranormed Space ◽  
Complex Sequences ◽  
Complex Linear ◽  
Topological Linear ◽  
Subadditive Function

A paranormed space X = (X, g) is a topological linear space in which the topology is given by paranorm g—a real subadditive function on X such that g(θ) = 0, g(x) = g(−x) and such that multiplication is continuous. In the above, θ is the zero in the complex linear space X and continuity of multiplication means that λn → λ, xn → x(i.e. g(xn − x) → 0) imply λnxn → λx, for scalars λ and vectors x. We shall use the term semimetric function to describe a real subadditive function g on X such that g(θ) = 0, g(x) = g(−x). Two familiar paranormed sequence spaces, which have been extensively studied (3), are l(p) and m(p). For a given sequence p = (gk) of strictly positive numbers, l;(p) is the set of all complex sequences x = (xk) such that and m(p) is the set of x such that sup Throughout, sums and suprema without limits are taken from 1 to ∞. Simons (3) considered only the case in which 0 < pk ≤ 1 so that natural paranorms would seem to be in m(p). In fact Simons showed that g1 was a paranorm for l(p), but that g2 did not satisfy the continuity of multiplication axiom.

On the convex kernel of a compact set

1967 ◽  
Vol 63 (2) ◽  
pp. 311-313 ◽  
Author(s):  
D. G. Larman
Keyword(s):  
Linear Space ◽  
Compact Subset ◽  
Line Segment ◽  
Linear Dimension ◽  
Single Point ◽  
Compact Set ◽  
Topological Linear Space ◽  
Topological Linear

Suppose that E is a compact subset of a topological linear space ℒ. Then the convex kernel K, of E, is such that a point k belongs to K if every point of E can be seen, via E, from k. Valentine (l) has asked for conditions on E which ensure that the convex kernel K, of E, consists of exactly one point, and in this note we give such a condition. If A, B, C are three subsets of E, we use (A, B, C) to denote the set of those points of E, which can be seen, via E, from a triad of points a, b, c, with a ∈ A, b ∈ B, c ∈ C. We shall say that E has the property if, whenever A is a line segment and B, C are points of E which are not collinear with any point of A, the set (A, B, C) has linear dimension of at most one, and degenerates to a single point whenever A is a point.

Continuous and Köthe–Toeplitz duals of certain sequence spaces

1969 ◽  
Vol 65 (2) ◽  
pp. 431-435 ◽  
Author(s):  
I. J. Maddox
Keyword(s):  
Sequence Spaces ◽  
Continuous Linear ◽  
Linear Functionals ◽  
Paranormed Space ◽  
Complex Sequences ◽  
Image Position

If (X, g) is a paranormed space, with paranorm g (see (2)), then we denote by X* the continuous dual of X, i.e. the set of all continuous linear functionals on X. If E is a set of complex sequences x = (xk) then E† will denote the generalized Köthe–Toeplitz dual of E

The graph-theoretic analogue of Tietze's characterization of a convex set and its generalization

1972 ◽  
Vol 72 (1) ◽  
pp. 7-9
Author(s):  
M. D. Guay
Keyword(s):  
Linear Space ◽  
Closed Subset ◽  
Convex Sets ◽  
Convex Set ◽  
Topological Linear Space ◽  
Connected Subset ◽  
Graph Theoretic ◽  
Topological Linear ◽  
Locally Convex

Introduction. One of the most satisfying theorems in the theory of convex sets states that a closed connected subset of a topological linear space which is locally convex is convex. This was first established in En by Tietze and was later extended by other authors (see (3)) to a topological linear space. A generalization of Tietze's theorem which appears in (2) shows if S is a closed subset of a topological linear space such that the set Q of points of local non-convexity of S is of cardinality n < ∞ and S ~ Q is connected, then S is the union of n + 1 or fewer convex sets. (The case n = 0 is Tietze's theorem.)

scholarly journals Matrix transformations and application to perturbed problems of some sequence spaces equations with operators

Filomat ◽  
2018 ◽  
Vol 32 (14) ◽  
pp. 5123-5130
Author(s):  
Malafosse de ◽  
Ali Fares ◽  
Ali Ayad
Keyword(s):  
Linear Space ◽  
Sequence Spaces ◽  
Matrix Transformations ◽  
Real Numbers ◽  
Positive Real ◽  
Complex Sequences ◽  
Perturbed Equation

Given any sequence z = (zn)n?1 of positive real numbers and any set E of complex sequences, we write Ez for the set of all sequences y = (yn)n?1 such that y/z = (yn/zn)n?1 ? E; in particular, cz = s(c) z denotes the set of all sequences y such that y/z converges. Starting with the equation Fx = Fb we deal with some perturbed equation of the form ? + Fx = Fb, where ? is a linear space of sequences. In this way we solve the previous equation where ? =(Ea)T and (E,F) ? {(l?,c), (c0,l?), (c0,c), (lp,c), (lp,l?), (w0,l?)} with p ? 1, and T is a triangle.

scholarly journals Matrix transformations and convex spaces

1988 ◽  
Vol 11 (3) ◽  
pp. 585-588
Author(s):  
I. J. Maddox
Keyword(s):  
Linear Space ◽  
Matrix Transformations ◽  
Topological Linear Space ◽  
Sequential Completeness ◽  
Metrizable Spaces ◽  
Topological Linear ◽  
Convex Spaces

In a Hausdorff topological linear space we examine relations betweenr-convexity and a condition on matrix transformations between null sequences. In particular, for metrizable spaces the condition impliesr-convexity. For locally bounded spaces the condition implies sequential completeness.

C(X) As A Dual Space

1972 ◽  
Vol 24 (3) ◽  
pp. 485-491 ◽  
Author(s):  
E. G. Manes
Keyword(s):  
Banach Space ◽  
Linear Space ◽  
Dual Space ◽  
Minkowski Functional ◽  
Topological Linear Space ◽  
Open Set ◽  
Topological Linear

It is known [1] that for compact Hausdorff X, C(X) is the dual of a Banach space if and only if X is hyperstonian, that is the closure of an open set in X is again open and the carriers of normal measures in C(X)* have dense union in X. With the desiratum of proving that C(X) is always the dual of some sort of space we broaden the concept of Banach space as follows. A Banach space may be comfortably regarded as a pair (E, B) where E is a topological linear space and B is a subset of E ; the requisite property is that the Minkowski functional of B be a complete norm whose topology coincides with that of E.

A Tauberian theorem for statistical convergence

1989 ◽  
Vol 106 (2) ◽  
pp. 277-280 ◽  
Author(s):  
I. J. Maddox
Keyword(s):  
Linear Space ◽  
Tauberian Theorem ◽  
Statistical Convergence ◽  
Slow Oscillation ◽  
Topological Linear Space ◽  
Tauberian Condition ◽  
Topological Linear ◽  
Locally Convex

The notion of statistical convergence of a sequence (xk) in a locally convex Hausdorff topological linear space X was introduced recently by Maddox[5], where it was shown that the slow oscillation of (sk) was a Tauberian condition for the statistical convergence of (sk).

Convex series

1972 ◽  
Vol 72 (1) ◽  
pp. 37-47 ◽  
Author(s):  
G. J. O. Jameson
Keyword(s):  
Linear Space ◽  
Topological Linear Space ◽  
Compact Sets ◽  
Topological Linear

1. Introduction. Let A be a subset of a Hausdorff topological linear space. By a convex series of elements of A we mean a series of the form where an∈A and λn ≥ 0 for each n, and . We say that A is:(i) CS-closed if it contains the sum of every convergent convex series of its elements;(ii) CS-compact if every convex series of its elements converges to a point of A (this bold terminology is chosen because sets satisfying this condition turn out to have properties analogous to those of compact sets).

On some new generalized difference sequence spaces on seminormed spaces defined by a sequence of Orlicz functions

2011 ◽  
Vol 61 (2) ◽  
Author(s):  
Çiğdem Bektaş
Keyword(s):  
Linear Space ◽  
Sequence Space ◽  
Sequence Spaces ◽  
Difference Sequence ◽  
Topological Properties ◽  
Orlicz Functions ◽  
Inclusion Relations ◽  
Complex Linear ◽  
Difference Sequence Spaces

AbstractIn this paper we define the sequence space ℓ M(Δυm, p, q, s) on a seminormed complex linear space, by using a sequence of Orlicz functions. We study some algebraic and topological properties. We prove some inclusion relations involving ℓ M(Δυm, p, q, s). spaces

Absolute convexity in certain topological linear spaces

1969 ◽  
Vol 66 (3) ◽  
pp. 541-545 ◽  
Author(s):  
I. J. Maddox ◽  
J. W. Roles
Keyword(s):  
Linear Space ◽  
Topological Linear Space ◽  
Linear Spaces ◽  
Empty Subset ◽  
Image Position ◽  
Topological Linear

For r > 0 a non-empty subset U of a linear space is said to be absolutely r-convex if x, y ∈ U and |λ|r + |μ|r ≤ 1 together imply λx + μy∈ U, or, equivalently, xl, …, xn∈ U andIt is clear that if U is absolutely r-convex, then it is absolutely s-convex whenever s < r. A topological linear space is said to be r-convex if every neighbourhood of the origin θ contains an absolutely r-convex neighbourhood of the origin. For the case 0 < r ≤ 1, these concepts were introduced and discussed by Landsberg(2).

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