The Bornological Tensor Product of two Riesz Spaces

A Riesz space tensor product of Archimedean Riesz spaces was introduced by D. H. Fremlin[2, 3]. His construction as well as a subsequent simplified version by H. H. Schaefer[10] depended on representation techniques and it is our aim to find a more direct way to prove the existence of the tensor product and to derive its properties. This tensor product proved to be extremely useful in the theory of positive operators on Banach lattices (see [3] and [10]) and should be considered as one of the basic constructions in the theory of Riesz spaces. It is therefore of interest to construct it in an intrinsic way. The problem to do this was already posed by Fremlin in [2]. In this paper we shall present two different approaches, the first of which is analogous to the formation of a free lattice generated by a given partially ordered set. (See [5], p. 41.) In the second one we first assume the Riesz spaces involved to have the principal projection property. In this case a simple method of construction by step-elements is available and the tensor product of arbitrary Archimedean Riesz spaces can then be obtained by embedding the spaces into their Dedekind completions. To complete the latter step we need results on the extension of Riesz bimorphisms which will be proved in §1. Both our approaches hinge on results about the tensor product of ordered vector spaces. It turns out that a unique tensor product for ordered vector spaces exists and is contained in the Riesz space tensor product. This is investigated in §2.

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AN f-ALGEBRA APPROACH TO THE RIESZ TENSOR PRODUCT OF ARCHIMEDEAN RIESZ SPACES

Quaestiones Mathematicae ◽

10.1080/16073606.1989.9632194 ◽

1989 ◽

Vol 12 (4) ◽

pp. 425-438 ◽

Cited By ~ 8

Author(s):

J J Grobler ◽

C C A Labuschagne

Keyword(s):

Tensor Product ◽

Riesz Spaces ◽

Algebra Approach

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Reference Signal Based Tensor Product Expansion for EOG-Related Artifact Separation in EEG

IEICE Transactions on Fundamentals of Electronics Communications and Computer Sciences ◽

10.1587/transfun.e100.a.2230 ◽

2017 ◽

Vol E100.A (11) ◽

pp. 2230-2237

Author(s):

Akitoshi ITAI ◽

Arao FUNASE ◽

Andrzej CICHOCKI ◽

Hiroshi YASUKAWA

Keyword(s):

Tensor Product ◽

Reference Signal

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On degeneracy problem of NFSRs via semi-tensor product

2020 39th Chinese Control Conference (CCC) ◽

10.23919/ccc50068.2020.9189105 ◽

2020 ◽

Author(s):

Xinyu Zhao ◽

Biao Wang ◽

Shuqian Zhu ◽

Jun-e Feng

Keyword(s):

Tensor Product ◽

Degeneracy Problem

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On the Buchstaber Subring in MSp∗

Georgian Mathematical Journal ◽

10.1515/gmj.1998.401 ◽

1998 ◽

Vol 5 (5) ◽

pp. 401-414

Author(s):

M. Bakuradze

Keyword(s):

Tensor Product ◽

Characteristic Classes ◽

Generalized Quaternion ◽

Symplectic Cobordisms

Abstract A formula is given to calculate the last n number of symplectic characteristic classes of the tensor product of the vector Spin(3)- and Sp(n)-bundles through its first 2n number of characteristic classes and through characteristic classes of Sp(n)-bundle. An application of this formula is given in symplectic cobordisms and in rings of symplectic cobordisms of generalized quaternion groups.

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Fractal Frames of Functions on the Rectangle

Fractal and Fractional ◽

10.3390/fractalfract5020042 ◽

2021 ◽

Vol 5 (2) ◽

pp. 42

Author(s):

María A. Navascués ◽

Ram Mohapatra ◽

Md. Nasim Akhtar

Keyword(s):

Tensor Product ◽

Product Space ◽

Integrable Function ◽

Two Dimensional ◽

Tensor Product Space ◽

Fractal Functions

In this paper, we define fractal bases and fractal frames of L2(I×J), where I and J are real compact intervals, in order to approximate two-dimensional square-integrable maps whose domain is a rectangle, using the identification of L2(I×J) with the tensor product space L2(I)⨂L2(J). First, we recall the procedure of constructing a fractal perturbation of a continuous or integrable function. Then, we define fractal frames and bases of L2(I×J) composed of product of such fractal functions. We also obtain weaker families as Bessel, Riesz and Schauder sequences for the same space. Additionally, we study some properties of the tensor product of the fractal operators associated with the maps corresponding to each variable.

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