scholarly journals An Adaptive Algorithm Employing Continuous Linear Functionals

2020 ◽  
pp. 161-181
Author(s):  
Yuhan Ding ◽  
Fred J. Hickernell ◽  
Lluís Antoni Jiménez Rugama
Keyword(s):  
Adaptive Algorithm ◽  
Continuous Linear ◽  
Linear Functionals

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.

scholarly journals Nonlinear potentials in function spaces

2002 ◽  
Vol 165 ◽  
pp. 91-116 ◽  
Author(s):  
Murali Rao ◽  
Zoran Vondraćek
Keyword(s):  
Banach Space ◽  
Potential Theory ◽  
Finite Energy ◽  
Duality Mapping ◽  
Quadratic Functional ◽  
Continuous Linear ◽  
Linear Functionals ◽  
Nonlinear Potential Theory ◽  
Nonlinear Potential ◽  
Nonlinear Potentials

We introduce a framework for a nonlinear potential theory without a kernel on a reflexive, strictly convex and smooth Banach space of functions. Nonlinear potentials are defined as images of nonnegative continuous linear functionals on that space under the duality mapping. We study potentials and reduced functions by using a variant of the Gauss-Frostman quadratic functional. The framework allows a development of other main concepts of nonlinear potential theory such as capacities, equilibrium potentials and measures of finite energy.

Linear Functionals and Summability Invariants

1974 ◽  
Vol 17 (2) ◽  
pp. 233-242 ◽  
Author(s):  
M. S. Macphail ◽  
A. Wilansky
Keyword(s):  
Continuous Linear ◽  
Linear Functionals ◽  
Convergence Domain

The purpose of this paper is to continue the study of certain “distinguished” subsets of the convergence domain of a matrix, as developed by A. Wilansky [6] and G. Bennett [1], We also consider continuous linear functionals on the domain, and the extent to which their representation is unique; this turns out to be connected with the behaviour of the subsets.

Left Cauchy Integral Bases in Linear Topological Spaces

1970 ◽  
Vol 13 (4) ◽  
pp. 431-439 ◽  
Author(s):  
James A. Dyer
Keyword(s):  
Linear Topological Space ◽  
Integral Representations ◽  
Topological Spaces ◽  
Continuous Linear ◽  
Linear Functionals ◽  
Natural Analogue ◽  
Cauchy Integral ◽  
Integral Basis ◽  
Integral Bases ◽  
Definition Of

The purpose of this paper is to consider a representation for the elements of a linear topological space in the form of a σ-integral over a linearly ordered subset of V; this ordered subset is what will be called an L basis. The formal definition of an L basis is essentially an abstraction from ideas used, often tacitly, in proofs of many of the theorems concerning integral representations for continuous linear functionals on function spaces.The L basis constructed in this paper differs in several basic ways from the integral basis considered by Edwards in [5]. Since the integrals used here are of Hellinger type rather than Radon type one has in the approximating sums for the integral an immediate and natural analogue to the partial sum operators of summation basis theory.

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

Neural Networks for Optimal Approximation of Smooth and Analytic Functions

1996 ◽  
Vol 8 (1) ◽  
pp. 164-177 ◽  
Author(s):  
H. N. Mhaskar
Keyword(s):  
Neural Networks ◽  
Target Function ◽  
Activation Function ◽  
Basis Functions ◽  
Optimal Order ◽  
Order Of Approximation ◽  
Continuous Linear ◽  
Linear Functionals ◽  
Principal Element ◽  
Hidden Layer

We prove that neural networks with a single hidden layer are capable of providing an optimal order of approximation for functions assumed to possess a given number of derivatives, if the activation function evaluated by each principal element satisfies certain technical conditions. Under these conditions, it is also possible to construct networks that provide a geometric order of approximation for analytic target functions. The permissible activation functions include the squashing function (1 − e−x)−1 as well as a variety of radial basis functions. Our proofs are constructive. The weights and thresholds of our networks are chosen independently of the target function; we give explicit formulas for the coefficients as simple, continuous, linear functionals of the target function.

scholarly journals A Riesz representation theorem for cone-valued functions

1999 ◽  
Vol 4 (4) ◽  
pp. 209-229
Author(s):  
Walter Roth
Keyword(s):  
Hausdorff Space ◽  
Inductive Limit ◽  
Riesz Representation Theorem ◽  
Continuous Linear ◽  
Linear Functionals ◽  
Borel Measures ◽  
Riesz Representation ◽  
Inductive Limit Topology ◽  
Locally Compact Hausdorff Space ◽  
Locally Convex

We consider Borel measures on a locally compact Hausdorff space whose values are linear functionals on a locally convex cone. We define integrals for cone-valued functions and verify that continuous linear functionals on certain spaces of continuous cone-valued functions endowed with an inductive limit topology may be represented by such integrals.

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