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

scholarly journals Curves with zero derivative in F-spaces

1981 ◽  
Vol 22 (1) ◽  
pp. 19-29 ◽  
Author(s):  
N. J. Kalton
Keyword(s):  
Characteristic Function ◽  
Mean Value ◽  
Mean Value Theorem ◽  
Continuous Linear ◽  
Linear Functionals ◽  
Continuous Map ◽  
Left And Right ◽  
The Mean ◽  
Image Position ◽  
Locally Convex

Let X be an F-space (complete metric linear space) and suppose g:[0, 1] → X is a continuous map. Suppose that g has zero derivative on [0, 1], i.e.for 0≤t≤1 (we take the left and right derivatives at the end points). Then, if X is locally convex or even if it merely possesses a separating family of continuous linear functionals, we can conclude that g is constant by using the Mean Value Theorem. If however X* = {0} then it may happen that g is not constant; for example, let X = Lp(0, 1) (0≤p≤1) and g(t) = l[0,t] (0≤t≤1) (the characteristic function of [0, t]). This example is due to Rolewicz [6], [7; p. 116].

Multipliers on some sequence spaces

1972 ◽  
Vol 72 (3) ◽  
pp. 393-401 ◽  
Author(s):  
J. C. Kurtz
Keyword(s):  
Normal Matrix ◽  
Sequence Spaces ◽  
The Matrix ◽  
Complex Sequences ◽  
Image Position

AbstractLet ω be the space of all (complex) sequences. If E, F are subspaces of ω and if A is any (infinite) normal matrix, we setandIf A is the matrix of a sequence–sequence summability transform, Ā and  shall denote the series–sequence and series–series forms of the transform, respectively. The multiplier spaces M(c(Ā), c(Ā)), M(lp(Â), l1(Â)) and M(lp(Â), c(Ā)) are characterized (1 ≤ p < ∞). Partial results are given for the spaces M(c(Ā), lp(Â)) and M(lp(Â), lp(Â)).

scholarly journals Linear functionals on Orlicz sequence spaces without local convexity

1992 ◽  
Vol 15 (2) ◽  
pp. 241-254 ◽  
Author(s):  
Marian Nowak
Keyword(s):  
Sequence Space ◽  
Sequence Spaces ◽  
Local Convexity ◽  
Continuous Linear ◽  
Linear Functionals ◽  
Orlicz Sequence Space ◽  
Orlicz Sequence Spaces ◽  
Locally Convex

The general form of continuous linear functionals on an Orlicz sequence space1ϕ(non-separable and non-locally convex in general) is obtained. It is proved that the spacehϕis anM-ideal in1ϕ.

Paranormed sequence spaces generated by infinite matrices

1968 ◽  
Vol 64 (2) ◽  
pp. 335-340 ◽  
Author(s):  
I. J. Maddox
Keyword(s):  
Linear Space ◽  
Sequence Spaces ◽  
Topological Linear Space ◽  
Infinite Matrices ◽  
Paranormed Space ◽  
Complex Sequences ◽  
Complex Linear ◽  
Topological Linear ◽  
Subadditive Function

A paranormed space X = (X, g) is a topological linear space in which the topology is given by paranorm g—a real subadditive function on X such that g(θ) = 0, g(x) = g(−x) and such that multiplication is continuous. In the above, θ is the zero in the complex linear space X and continuity of multiplication means that λn → λ, xn → x(i.e. g(xn − x) → 0) imply λnxn → λx, for scalars λ and vectors x. We shall use the term semimetric function to describe a real subadditive function g on X such that g(θ) = 0, g(x) = g(−x). Two familiar paranormed sequence spaces, which have been extensively studied (3), are l(p) and m(p). For a given sequence p = (gk) of strictly positive numbers, l;(p) is the set of all complex sequences x = (xk) such that and m(p) is the set of x such that sup Throughout, sums and suprema without limits are taken from 1 to ∞. Simons (3) considered only the case in which 0 < pk ≤ 1 so that natural paranorms would seem to be in m(p). In fact Simons showed that g1 was a paranorm for l(p), but that g2 did not satisfy the continuity of multiplication axiom.

scholarly journals On the numerical radius of an element of a normed algebra

1974 ◽  
Vol 15 (1) ◽  
pp. 90-93
Author(s):  
GH. Mocanu
Keyword(s):  
Banach Space ◽  
Dual Space ◽  
Complex Field ◽  
Continuous Linear ◽  
Linear Functionals ◽  
Normed Algebra ◽  
Numerical Radius ◽  
Image Position

Let A be a unital normed algebra over the complex field ℂ, A' the dual space of A, i.e., the Banach space of all continuous linear functionals on A, and let S be the set of all states on A, i.e.,

Subalgebras of modular annihilator algebras

1969 ◽  
Vol 66 (1) ◽  
pp. 5-12 ◽  
Author(s):  
B. A. Barnes
Keyword(s):  
Banach Algebra ◽  
Linear Functional ◽  
Linear Functionals ◽  
Radical Algebra ◽  
Complex Functions ◽  
Closed Subalgebra ◽  
Point Evaluations ◽  
Complex Sequences ◽  
Simple Algebras ◽  
Image Position

Throughout this paper we deal only with complex and semi-simple algebras. Let B be such an algebra. We denote the socle of B as SB. B is a modular annihilator algebra if B/SB is a radical algebra, i.e. if every element of B is quasi-regular modulo the socle of B; see (1) or (12). Now assume that B is a modular annihilator algebra and a Banach algebra. Then any semi-simple closed subalgebra of B is a modular annihilator algebra by ((4), Cor. to Theorem 4·2,). It is not true, however, that any semi-simple subalgebra A of B is a modular annihilator algebra, even when A is a Banach algebra in some norm. We give a simple example to illustrate this. Let A be the algebra of all complex functions f, continuous on the closed unit disk D in the complex plane, analytic in the interior of D, and such that f(0) = 0. A is a Banach algebra in the usual sup norm over D. Now consider the norm on A defined byLet B be the completion of A in this norm. A has an involution * defined by and also ‖ff*‖ = ‖f‖2 for all f ∈ A. Therefore B is a B*-algebra. It is not difficult to verify that the only non-zero multiplicative linear functionals on A which are continuous with respect to the norm ‖·‖, are the point evaluations at 1/n, n = 1, 2 …. It follows that every non-zero multiplicative linear functional on B is an extension of one of these point evaluations to B. Thus B can be identified with the algebra of all complex sequences which converge to zero.

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.

Generalized Resolvent Equations and Unsymmetric Dirichlet Spaces

1981 ◽  
Vol 33 (5) ◽  
pp. 1111-1141
Author(s):  
Joanne Elliott
Keyword(s):  
Linear Operators ◽  
Dirichlet Spaces ◽  
Resolvent Equation ◽  
Continuous Linear ◽  
Generalized Resolvent ◽  
Resolvent Equations ◽  
Measure Spaces ◽  
Image Position ◽  
Generalized Resolvent Equations ◽  
Continuous Linear Operators

Let (X, , μ) and (X, , μ′) be measure spaces with the measures μ and μ′ totally finite. Suppose {Uλ: λ > 0} is a family of positive (i.e., ϕ ≧ 0 ⇒ Uλϕ ≧ 0) continuous linear operators from L2(X, dμ′) to L2(X,dμ) with the following additional properties: if ϕ ≧ 0 then Uλϕ is non-decreasing as λ increases, while λ−1Uλϕ is nonincreasing.A family {Mλ:λ > 0} of continuous linear operators from L2(X, dμ) to L2(X, dμ′) satisfies the “generalized resolvent equation” relative to {Uλ} if(0.1)for positive λ and v. If Uλ = λI, then (0.1) is just the well-known resolvent equation. The family {Mλ} is called submarkov if Mλ is a positive operator and(0.2)it is conservative if(0.3)

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