Some Multiplicative Functionals

1953 ◽  
Vol 5 ◽  
pp. 174-178 ◽  
Author(s):  
D. G. Bourgin
Keyword(s):  
Continuous Function ◽  
Compact Space ◽  
Banach Algebras ◽  
The Real ◽  
Function Ring ◽  
Multiplicative Functionals

This note concerns itself primarily with the representation of continuous multiplicative functionals on L2 types of rings or Banach algebras to the real or complex fields where convolution is taken as the ring multiplication. In a recent publication [1] such functionals were studied for the continuous function ring C(S) over a compact space S.

scholarly journals Wirtinger integral inequalities for pseudo-integrals and pseudo-additive measure

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Jing Guo ◽  
Xianjun Zhu
Keyword(s):  
Continuous Function ◽  
Integral Inequalities ◽  
Additive Measure ◽  
The Real ◽  
First Case ◽  
Interval Valued

AbstractThe main purpose of this paper is to show Wirtinger type inequalities for the pseudo-integral. We are concerned with pseudo-integrals based on the following three canonical cases: in the first case, the real semiring with pseudo-operation is generated by a strictly monotone continuous function g; in the second case, the pseudo-operations include a pseudo-multiplication and a power arithmetic addition; in the last case, ⊕-measures are interval-valued. Examples are given to illustrate these equalities.

A Corson Compact Space is Countable if the Complement of its Diagonal is Functionally Countable

2021 ◽  
Vol 58 (3) ◽  
pp. 398-407
Author(s):  
Vladimir V. Tkachuk
Keyword(s):  
Continuous Function ◽  
Compact Space ◽  
Infinite Cardinal ◽  
Scattered Space ◽  
Corson Compact

A space X is called functionally countable if ƒ (X) is countable for any continuous function ƒ : X → Ø. Given an infinite cardinal k, we prove that a compact scattered space K with d(K) > k must have a convergent k+-sequence. This result implies that a Corson compact space K is countable if the space (K × K) \ ΔK is functionally countable; here ΔK = {(x, x): x ϵ K} is the diagonal of K. We also establish that, under Jensen’s Axiom ♦, there exists a compact hereditarily separable non-metrizable compact space X such that (X × X) \ ΔX is functionally countable and show in ZFC that there exists a non-separable σ-compact space X such that (X × X) \ ΔX is functionally countable.

On Certain Extensions of Function Rings

1959 ◽  
Vol 11 ◽  
pp. 87-96
Author(s):  
Bernhard Banaschewski
Keyword(s):  
Topological Space ◽  
Galois Extension ◽  
Present Note ◽  
Banach Algebras ◽  
Faithful Representation ◽  
Normal Extension ◽  
Invariance Group ◽  
Group Of Automorphisms ◽  
Normed Ring ◽  
Function Ring

The present note is concerned with the existence and properties of certain types of extensions of Banach algebras which allow a faithful representation as the normed ring C(E) of all bounded continuous real functions on some topological space E. These Banach algebras can be characterized intrinsically in various ways (1); they will be called function rings here. A function ring E will be called a normal extension of a function ring G if E is directly indecomposable, contains C as a Banach subalgebra and possesses a group G of automorphisms for which C is the ring of invariants, that is, the set of all elements fixed under G. G will then be called a group of automorphisms of E over C. If E is a normal extension of C with precisely one group of automorphisms over C, which is then the invariance group of C in E, then E will be called a Galois extension of C. Such an extension will be called finite if its group is finite.

Gleason Parts of Real Function Algebras

1981 ◽  
Vol 33 (1) ◽  
pp. 181-200 ◽  
Author(s):  
S. H. Kulkarni ◽  
B. V. Limaye
Keyword(s):  
Real Function ◽  
Banach Algebras ◽  
Complex Function ◽  
Klein Surfaces ◽  
The Real ◽  
Uniform Algebras ◽  
Function Algebras ◽  
Complex Valued ◽  
Systematic Exposition

Although the theory of complex Banach algebras is by now classical, the first systematic exposition of the theory of real Banach algebras was given by Ingelstam [5] as late as 1965. More recently, further attention to real Banach algebras was paid in 1970 [1], where, among other things, the (real) standard algebras on finite open Klein surfaces were introduced. Generalizing these considerations, real uniform algebras were studied in [7] and [6].In the present paper, an attempt is made to develop the theory of real function algebras (see Section 1 for the definition) along the lines of the complex function algebras. Although the real function algebras are not structurally different from the real uniform algebras introduced in [7], they are easier to deal with since their elements are actually (complex-valued) functions.

scholarly journals A Remark on Isometries of Absolutely Continuous Spaces

2020 ◽  
Vol 2020 ◽  
pp. 1-3
Author(s):  
Alireza Ranjbar-Motlagh
Keyword(s):  
Continuous Function ◽  
Composition Operator ◽  
Real Line ◽  
Weighted Composition Operator ◽  
Function Spaces ◽  
Absolutely Continuous ◽  
The Real ◽  
Absolutely Continuous Function ◽  
Weighted Composition ◽  
Vector Valued

The purpose of this article is to study the isometries between vector-valued absolutely continuous function spaces, over compact subsets of the real line. Indeed, under certain conditions, it is shown that such isometries can be represented as a weighted composition operator.

Boundaries For Real Banach Algebras

1976 ◽  
Vol 28 (1) ◽  
pp. 42-49 ◽  
Author(s):  
B. V. Limaye
Keyword(s):  
Banach Algebra ◽  
Maximal Ideal ◽  
Banach Algebras ◽  
Maximal Ideal Space ◽  
Close Connection ◽  
Ideal Space ◽  
The Real ◽  
Šilov Boundary ◽  
The Absolute ◽  
Silov Boundary

Let A be a commutative real Banach algebra with unit, and MA its maximal ideal space. The existence of the Silov boundary SA for A was established in [5] by resorting to the complexification of A. We give here an intrinsic proof of this result which exhibits the close connection between the absolute values and the real parts of ‘functions’ in A (Theorem 1.3).

Maximal ideals of Banach algebras of measures on the real line

1978 ◽  
Vol 24 (3) ◽  
pp. 669-671
Author(s):  
B. A. Rogizin ◽  
M. S. Sgibnev
Keyword(s):  
Real Line ◽  
Banach Algebras ◽  
The Real ◽  
Maximal Ideals

scholarly journals A Single Hidden Layer Feedforward Network with Only One Neuron in the Hidden Layer Can Approximate Any Univariate Function

2016 ◽  
Vol 28 (7) ◽  
pp. 1289-1304 ◽  
Author(s):  
Namig J. Guliyev ◽  
Vugar E. Ismailov
Keyword(s):  
Continuous Function ◽  
Finite Interval ◽  
Activation Function ◽  
Feedforward Network ◽  
Arbitrary Continuous Function ◽  
The Real ◽  
Univariate Function ◽  
Constructive Approximation ◽  
Hidden Layer ◽  
Sigmoidal Activation Function

The possibility of approximating a continuous function on a compact subset of the real line by a feedforward single hidden layer neural network with a sigmoidal activation function has been studied in many papers. Such networks can approximate an arbitrary continuous function provided that an unlimited number of neurons in a hidden layer is permitted. In this note, we consider constructive approximation on any finite interval of [Formula: see text] by neural networks with only one neuron in the hidden layer. We construct algorithmically a smooth, sigmoidal, almost monotone activation function [Formula: see text] providing approximation to an arbitrary continuous function within any degree of accuracy. This algorithm is implemented in a computer program, which computes the value of [Formula: see text] at any reasonable point of the real axis.

scholarly journals Continuous functions of bounded nth variation

1969 ◽  
Vol 16 (3) ◽  
pp. 205-214
Author(s):  
Gavin Brown
Keyword(s):  
Continuous Function ◽  
Positive Integer ◽  
Decomposition Theorem ◽  
Continuous Functions ◽  
Unit Interval ◽  
Jordan Decomposition ◽  
The Real ◽  
Elementary Construction ◽  
The Difference

Let n be a positive integer. We give an elementary construction for the nth variation, Vn(f), of a real valued continuous function f and prove an analogue of the classical Jordan decomposition theorem. In fact, let C[0, 1] denote the real valued continuous functions on the closed unit interval, let An denote the semi-algebra of non-negative functions in C[0, 1] whose first n differences are non-negative, and let Sn denote the difference algebra An - An. We show that Sn is precisely that subset of C[0, 1] on which Vn(f)<∞. (Theorem 1).

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