Functional continuity of topological algebras with orthonormal bases

2020 ◽  
pp. 2150059
Author(s):  
G. Siva ◽  
C. Ganesa Moorthy
Keyword(s):  
Orthonormal Basis ◽  
Orthogonal Basis ◽  
Linear Functionals ◽  
Topological Algebras ◽  
Orthonormal Bases ◽  
Commutative Algebras ◽  
Positive Linear Functionals ◽  
Functional Value ◽  
Locally Convex ◽  
Locally Convex Algebra

(i) Every complex [Formula: see text]-algebra with an identity and with an orthonormal basis is functionally continuous; (ii) Every complex complete LMC algebra with an orthogonal basis is functionally continuous; (iii) Every complex sequentially complete locally convex algebra with an unconditional orthonormal basis and with an element [Formula: see text] for which [Formula: see text]th coefficient functional value tends to infinity as [Formula: see text] tends to infinity is functionally continuous. These results are proved and an example is provided for non-extendability of these results. A representation for positive linear functionals on a sequentially complete locally convex algebra with an unconditional orthonormal basis, with an identity, and with an element [Formula: see text] mentioned in (iii) is obtained. All results are obtained only for commutative algebras.

Infrasequential Topological Algebras

1979 ◽  
Vol 22 (4) ◽  
pp. 413-418 ◽  
Author(s):  
T. Husain
Keyword(s):  
Linear Functional ◽  
Topological Algebra ◽  
Topological Algebras ◽  
Locally Convex ◽  
Locally Convex Algebra

The notion of sequential topological algebra was introduced by this author and Ng [3], Among a number of results concerning these algebras, we showed that each multiplicative linear functional on a sequentially complete, sequential, locally convex algebra is bounded ([3], Theorem 1). From this it follows that every multiplicative linear functional on a sequential F-algebra (complete metrizable) is continuous ([3], Corollary 2).

scholarly journals A note on order topologies on ordered tensor products

1970 ◽  
Vol 3 (3) ◽  
pp. 385-390
Author(s):  
G. Davis
Keyword(s):  
Tensor Product ◽  
Uniform Convergence ◽  
Tensor Products ◽  
Order Topology ◽  
Vector Spaces ◽  
Locally Convex Spaces ◽  
Linear Functionals ◽  
Positive Linear Functionals ◽  
Ordered Vector Spaces ◽  
Locally Convex

If E, F are regularly ordered vector spaces the tensor product E ⊗ F can be ordered by the conic hull Kπ of tensors, x ⊗ y with x ≥ 0 in E and y ≥ 0 in F, or by the cone K⊗ of tensors Φ ∈ E ⊗ F such that Φ(x′, y′) ≥ 0 for positive linear functionals x′, y′ on E, F.If E, F are locally convex spaces the tensor product can te given the π-topology which is defined by seminorms pα ⊕ qβ where {pα}, {qβ} are classes of seminorms defining the topologies on E, F. The tensor product can also be given the ε-topology which is the topology of uniform convergence on equicontinuous subsets J x H of E′ x F′. The main result of this note is that if the regularly ordered vector spaces E, F carry their order topologies then the order topology on E ⊕ F is the π-topology when E ⊕ F is ordered by kπ, and the ε-topology when E ⊕ F is ordered by K⊕.

scholarly journals Multiplicative functionals and a class of topological algebras

1977 ◽  
Vol 17 (3) ◽  
pp. 391-399 ◽  
Author(s):  
Gerard A. Joseph
Keyword(s):  
Linear Functional ◽  
Topological Algebra ◽  
Null Sequence ◽  
Topological Algebras ◽  
Multiplicative Functionals ◽  
Locally Convex ◽  
Locally Convex Algebra

Every multiplicative linear functional on a pseudocomplete locally convex algebra satisfying the “sequential” property of Husain and Ng is bounded (a topological algebra is called “sequential” if every null sequence contains an element whose powers converge to zero). Characterizations of such algebras are given, with some examples.

scholarly journals Extending Edgar's ordering to locally convex spaces

1992 ◽  
Vol 34 (2) ◽  
pp. 175-188
Author(s):  
Neill Robertson
Keyword(s):  
Convex Space ◽  
Dual Pair ◽  
Locally Convex Space ◽  
Vector Spaces ◽  
Locally Convex Spaces ◽  
Continuous Linear ◽  
Linear Functionals ◽  
Hausdorff Topological Vector Space ◽  
Locally Convex ◽  
Mackey Topologies

By the term “locally convex space”, we mean a locally convex Hausdorff topological vector space (see [17]). We shall denote the algebraic dual of a locally convex space E by E*, and its topological dual by E′. It is convenient to think of the elements of E as being linear functionals on E′, so that E can be identified with a subspace of E′*. The adjoint of a continuous linear map T:E→F will be denoted by T′:F′→E′. If 〈E, F〈 is a dual pair of vector spaces, then we shall denote the corresponding weak, strong and Mackey topologies on E by α(E, F), β(E, F) and μ(E, F) respectively.

Interval oscillation criteria for self-adjoint matrix Hamiltonian systems

2005 ◽  
Vol 135 (5) ◽  
pp. 1085-1108 ◽  
Author(s):  
Qigui Yang ◽  
Yun Tang
Keyword(s):  
Hamiltonian Systems ◽  
Half Line ◽  
Linear Functionals ◽  
Matrix Space ◽  
Differential Systems ◽  
Adjoint Matrix ◽  
Oscillation Criteria ◽  
Positive Linear Functionals ◽  
Interval Oscillation ◽  
New Criteria

By using a monotonic functional on a suitable matrix space, some new oscillation criteria for self-adjoint matrix Hamiltonian systems are obtained. They are different from most known results in the sense that the results of this paper are based on information only for a sequence of subintervals of [t0, ∞), rather than for the whole half-line. We develop new criteria for oscillations involving monotonic functionals instead of positive linear functionals or the largest eigenvalue. The results are new, even for the particular case of self-adjoint second-differential systems which can be applied to extreme cases such as

scholarly journals A class of factorable topological algebras

1990 ◽  
Vol 33 (1) ◽  
pp. 53-59 ◽  
Author(s):  
E. Ansari-Piri
Keyword(s):  
Vector Space ◽  
Topological Vector Space ◽  
Approximate Identity ◽  
Necessary Condition ◽  
Topological Algebras ◽  
Approximate Identities ◽  
Cohen Factorization ◽  
Locally Convex ◽  
Space Property ◽  
Uniformly Bounded

The famous Cohen factorization theorem, which says that every Banach algebra with bounded approximate identity factors, has already been generalized to locally convex algebras with what may be termed “uniformly bounded approximate identities”. Here we introduce a new notion, that of fundamentality generalizing both local boundedness and local convexity, and we show that a fundamental Fréchet algebra with uniformly bounded approximate identity factors. Fundamentality is a topological vector space property rather than an algebra property. We exhibit some non-fundamental topological vector space and give a necessary condition for Orlicz space to be fundamental.

Orthonormal basis of wavelets with customizable frequency bands

2016 ◽  
Vol 14 (06) ◽  
pp. 1650050 ◽  
Author(s):  
Hiroshi Toda ◽  
Zhong Zhang
Keyword(s):  
Frequency Domain ◽  
Infinite Number ◽  
Orthonormal Basis ◽  
The Other ◽  
Just Intonation ◽  
Frequency Bands ◽  
Orthonormal Bases ◽  
Major Scale ◽  
Dilation Factor ◽  
New Type

We already proved the existence of an orthonormal basis of wavelets having an irrational dilation factor with an infinite number of wavelet shapes, and based on its theory, we proposed an orthonormal basis of wavelets with an arbitrary real dilation factor. In this paper, with the development of these fundamentals, we propose a new type of orthonormal basis of wavelets with customizable frequency bands. Its frequency bands can be freely designed with arbitrary bounds in the frequency domain. For example, we show two types of orthonormal bases of wavelets. One of them has an irrational dilation factor, and the other is designed based on the major scale in just intonation.

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