A simple proof of the Mazur-Orlicz summability theorem

1981 ◽  
Vol 89 (3) ◽  
pp. 391-392 ◽  
Author(s):  
J. A. Fridy
Keyword(s):  
Functional Analysis ◽  
Simple Proof ◽  
Direct Proof ◽  
Topological Properties ◽  
Unbounded Sequence ◽  
Important Theorem ◽  
Slowly Oscillating Sequences

In 1933(1), Mazur and Orlicz stated that, if a conservative (i.e. convergence preserving) matrix sums a bounded nonconvergent sequence, then it must sum an unbounded sequence. Their proof of this result (2) was one of the early applications of functional analysis to summability theory, and it is based on rather deep topological properties of F K-spaces. In (3) Zeller obtained a proof of this important theorem as a consequence of his study of the summability of slowly oscillating sequences. The purpose of this note is to give a simple direct proof of this theorem using only the well-known Silverman-Töplitz conditions for regularity. In order to reduce the details of the argument, we state and prove the result for regular matrices rather than conservative matrices.

scholarly journals A Simple Proof of the Jamiołkowski Criterion for Complete Positivity of Linear Maps

2005 ◽  
Vol 12 (01) ◽  
pp. 55-64 ◽  
Author(s):  
D. Salgado ◽  
J. L. Sánchez-Gómez ◽  
M. Ferrero
Keyword(s):  
Simple Proof ◽  
Direct Proof ◽  
Complete Positivity ◽  
Systematic Method ◽  
Completely Positive ◽  
Linear Map ◽  
Matrix Algebras ◽  
Linear Maps ◽  
Kraus Decomposition

We give a simple direct proof of the Jamiołkowski criterion to check whether a linear map between matrix algebras is completely positive or not. This proof is more accessible for physicists than other ones found in the literature and provides a systematic method to give any set of Kraus matrices of the Kraus decomposition.

scholarly journals On Certain Topological Properties of Normed Space Valued Null Function Space c0 (S, (E, || . || ), xi, u) and Its Paranormed Structure

2015 ◽  
Vol 15 (1) ◽  
pp. 121-128
Author(s):  
Narayan Prasad Pahari
Keyword(s):  
Functional Analysis ◽  
Function Space ◽  
Normed Space ◽  
Function Spaces ◽  
Sequence Spaces ◽  
Basic Sequence ◽  
Topological Properties ◽  
New Class ◽  
And Function ◽  
Class Of Functions

The aim of this paper is to introduce and study a new class c0 (S, (E, || . || ), ξ, u) of normed space E valued functions which will generalize some of the well known  basic sequence spaces and function spaces studied in Functional Analysis.. Beside the investigation pertaining to the linear paranormed structure of the class c0 ( S, (E, || . || ), ξ, u ) when topologized it with suitable natural paranorm , our primarily interest is to explore the conditions pertaining the containment relation of the class c0 (S, (E, || . || ), ξ, u) in terms of different ξ and u so that such a class of functions is contained in or equal to another class of similar nature.DOI: http://dx.doi.org/10.3126/njst.v15i1.12028Nepal Journal of Science and TechnologyVol. 15, No.1 (2014) 121-128

scholarly journals Estimates of (1+x)1/x involved in Carleman’s inequality and Keller’s limit

Filomat ◽  
2015 ◽  
Vol 29 (7) ◽  
pp. 1535-1539 ◽  
Author(s):  
Cristinel Mortici ◽  
X.J. Jang
Keyword(s):  
Simple Proof ◽  
Direct Proof ◽  
Constant E ◽  
Carleman’S Inequality ◽  
Keller’S Limit ◽  
Carleman's Inequality

The aim of this work is to extend the results obtained by Yang Bicheng and L. Debnath in [Some inequalities involving the constant e and an application to Carleman?s inequality, J. Math. Anal. Appl., 223 (1998), 347-353]. We present a simple proof of our new result which can be also used as a direct proof for Yang Bicheng and L. Debnath results. Finally some applications to generalized Keller?s limit and further directions are provided.

scholarly journals On torus homeomorphisms semiconjugate to irrational rotations

2014 ◽  
Vol 35 (7) ◽  
pp. 2114-2137 ◽  
Author(s):  
T. JÄGER ◽  
A. PASSEGGI
Keyword(s):  
Simple Proof ◽  
Irrational Rotation ◽  
Rotation Vector ◽  
Topological Properties ◽  
Empty Interior ◽  
Irrational Rotations ◽  
Skew Products ◽  
Invariant Foliation ◽  
Special Case

In the context of the Franks–Misiurewicz conjecture, we study homeomorphisms of the two-torus semiconjugate to an irrational rotation of the circle. As a special case, this conjecture asserts uniqueness of the rotation vector in this class of systems. We first characterize these maps by the existence of an invariant ‘foliation’ by essential annular continua (essential subcontinua of the torus whose complement is an open annulus) which are permuted with irrational combinatorics. This result places the considered class close to skew products over irrational rotations. Generalizing a well-known result of Herman on forced circle homeomorphisms, we provide a criterion, in terms of topological properties of the annular continua, for the uniqueness of the rotation vector. As a byproduct, we obtain a simple proof for the uniqueness of the rotation vector on decomposable invariant annular continua with empty interior. In addition, we collect a number of observations on the topology and rotation intervals of invariant annular continua with empty interior.

Note on Centrobaric Shells

1897 ◽  
Vol 21 ◽  
pp. 117-118
Author(s):  
Tait
Keyword(s):  
Spherical Shell ◽  
Simple Proof ◽  
Surface Density ◽  
Natural Philosophy ◽  
Direct Proof ◽  
The Other ◽  
Internal Point ◽  
Other Hand ◽  
Internal Particle ◽  
As If

It is singular to observe the comparative ease with which elementary propositions in attraction can be proved by one of the obvious methods, while the proof by the other is tedious.Thus nothing can be simpler than Newton's proof that a uniform spherical shell exerts no gravitating force on an internal particle. But, so far as I know, there is no such simple proof (of a direct character) that the potential is constant throughout the interior.On the other hand the direct proof that a spherical shell, whose surface-density is inversely as the cube of the distance from an internal point, is centrobaric is neither short nor simple. (See, for instance, Thomson and Tait's Elements of Natural Philosophy, § 491.) But we may prove at once that its potential at external points is the same as if its mass were condensed at the internal point.

scholarly journals On a Boolean algebra of projections constructed by Dieudonné

1969 ◽  
Vol 16 (3) ◽  
pp. 259-262 ◽  
Author(s):  
H. R. Dowson
Keyword(s):  
Banach Space ◽  
Boolean Algebra ◽  
Simple Proof ◽  
Direct Proof ◽  
Complete Boolean Algebra ◽  
Direct Sum ◽  
Algebra Of Operators

Dieudonné (4) has constructed an example of a Banach space X and a complete Boolean algebra B̃ of projections on X such that B̃ has uniform multiplicity two, but for no choice of x1, x2 in X and non-zero E in B̃ is EX the direct sum of the cyclic subspaces clm {Ex1:E∈B̃} and clm {Ex2:E∈B̃}. Tzafriri observed that it could be deduced from Corollary 4 (9, p. 221) that the commutant B̃′ of B̃ is equal to A(B̃), the algebra of operators generated by B̃ in the uniform operator topology. A study of (3) suggested the direct proof of the second property given in this note. From this there follows a simple proof that B̃ has the first property.

scholarly journals Quasi-pseudometric properties of the Nikodym-Saks space

2003 ◽  
Vol 4 (2) ◽  
pp. 243 ◽  
Author(s):  
Jesús Ferrer
Keyword(s):  
Functional Analysis ◽  
Metric Space ◽  
Measure Theory ◽  
Complete Metric Space ◽  
Additive Measure ◽  
Topological Properties ◽  
Pseudometric Space

<p>For a non-negative finite countably additive measure μ defined on the σ-field Σ of subsets of Ω, it is well known that a certain quotient of Σ can be turned into a complete metric space Σ (Ω), known as the Nikodym-Saks space, which yields such important results in Measure Theory and Functional Analysis as Vitali-Hahn-Saks and Nikodym's theorems. Here we study some topological properties of Σ (Ω) regarded as a quasi-pseudometric space.</p>

scholarly journals About the primitivity theorem

1992 ◽  
Vol 25 (5) ◽  
pp. 661-662
Author(s):  
Y. Le Page
Keyword(s):  
Undergraduate Students ◽  
Basic Property ◽  
Simple Proof ◽  
Nauk Sssr ◽  
Three Dimensional ◽  
Four Dimensions ◽  
Important Theorem ◽  
Akad Nauk Sssr ◽  
A Cell ◽  
Selection Of

Loosely stated, the primitivity theorem says that a cell based on the three shortest noncoplanar translations of a lattice is primitive. No course on elementary crystallography can omit this basic property of three-dimensional lattices, with everyday applications for selection of cells and for cell reduction. Textbooks have treated this property as obvious for many years now and have not hinted at a proof. The complexity of several apparently little-known proofs published since 1831 and the fact that no similar theorem exists in four dimensions or more show that this property cannot be taken for granted. However, little more than a drawing schematizing the simple proof of Delaunay, Galiulin, Dolbilin, Zalgaller & Stogrin [Dokl. Akad. Nauk SSSR (1973), 209, 25–58] would be needed to clarify this important theorem for average undergraduate students.

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