scholarly journals A new minimax theorem and a perturbed James's theorem

2002 ◽  
Vol 66 (1) ◽  
pp. 43-56 ◽  
Author(s):  
M. Ruiz Galán ◽  
S. Simons
Keyword(s):  
Bilinear Form ◽  
Convex Subset ◽  
Direct Proof ◽  
Minimax Theorem ◽  
Locally Convex Space ◽  
Sufficient Condition ◽  
Nonempty Convex Subset ◽  
Nonempty Convex ◽  
Locally Convex ◽  
Special Case

The main result of this paper is a sufficient condition for the minimax relation to hold for the canonical bilinear form on X × Y, where X is a nonempty convex subset of a real locally convex space and Y is a nonempty convex subset of its dual. Using the known “converse minimax theorem”, this result leads easily to a nonlinear generalisation of James's (“sup”) theorem. We give a brief discussion of the connections with the “sup-limsup theorem” and, in the appendix to the paper, we give a simple, direct proof (using Goldstine's theorem) of the converse minimax theorem referred to above, valid for the special case of a normed space.

scholarly journals An application of KKM-map principle

1992 ◽  
Vol 15 (4) ◽  
pp. 659-661 ◽  
Author(s):  
A. Carbone
Keyword(s):  
Convex Subset ◽  
Fixed Point Theorems ◽  
Normed Linear Space ◽  
Compact Convex ◽  
Nonempty Convex Subset ◽  
Nonempty Convex ◽  
Coincidence Theorems ◽  
Nonempty Compact ◽  
Nonempty Compact Convex Subset ◽  
Kkm Map

The following theorem is proved and several fixed point theorems and coincidence theorems are derived as corollaries. LetCbe a nonempty convex subset of a normed linear spaceX,f:C→Xa continuous function,g:C→Ccontinuous, onto and almost quasi-convex. Assume thatChas a nonempty compact convex subsetDsuch that the setA={y∈C:‖g(x)−f(y)‖≥‖g(y)−f(y)‖   for   all   x∈D}is compact.Then there is a pointy0∈Csuch that‖g(y0)−f(y0)‖=d(f(y0),C).

scholarly journals Properties of the solutions of those equations for which the Krasnoselskii iteration converges

2012 ◽  
Vol 28 (2) ◽  
pp. 329-336
Author(s):  
IOAN A. RUS ◽  
Keyword(s):  
Banach Space ◽  
Fixed Points ◽  
Convex Subset ◽  
Data Dependence ◽  
Picard Operator ◽  
Nonempty Convex Subset ◽  
Krasnoselskii Iteration ◽  
Nonempty Convex ◽  
Weakly Picard Operator

Let (X, +, R, →) be a vectorial L-space, Y ⊂ X a nonempty convex subset of X and f : Y → Y be an operator with Ff := {x ∈ Y | f(x) = x} 6= ∅. Let 0 < λ < 1 and let fλ be the Krasnoselskii operator corresponding to f, i.e., fλ(x) := (1 − λ)x + λf(x), x ∈ Y. We suppose that fλ is a weakly Picard operator (see I. A. Rus, Picard operators and applications, Sc. Math. Japonicae, 58 (2003), No. 1, 191-219). The aim of this paper is to study some properties of the fixed points of the operator f: Gronwall lemmas and comparison lemmas (when (X, +, R, →, ≤) is an ordered L-space) and data dependence (when X is a Banach space). Some applications are also given.

scholarly journals Global Schauder decompositions of locally convex spaces

2007 ◽  
Vol 101 (1) ◽  
pp. 65
Author(s):  
Milena Venkova
Keyword(s):  
Convex Space ◽  
Entire Functions ◽  
Locally Convex Space ◽  
Locally Convex Spaces ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Necessary And Sufficient ◽  
Locally Convex ◽  
Convex Spaces

We define global Schauder decompositions of locally convex spaces and prove a necessary and sufficient condition for two spaces with global Schauder decompositions to be isomorphic. These results are applied to spaces of entire functions on a locally convex space.

scholarly journals Subordination for sequentially equicontinuous equibounded $$C_0$$-semigroups

2021 ◽  
Author(s):  
Karsten Kruse ◽  
Jan Meichsner ◽  
Christian Seifert
Keyword(s):  
Convex Space ◽  
Locally Convex Space ◽  
Bernstein Function ◽  
Fractional Powers ◽  
The One ◽  
Locally Convex ◽  
Special Case

AbstractWe consider operators A on a sequentially complete Hausdorff locally convex space X such that $$-A$$ - A generates a (sequentially) equicontinuous equibounded $$C_0$$ C 0 -semigroup. For every Bernstein function f we show that $$-f(A)$$ - f ( A ) generates a semigroup which is of the same ‘kind’ as the one generated by $$-A$$ - A . As a special case we obtain that fractional powers $$-A^{\alpha }$$ - A α , where $$\alpha \in (0,1)$$ α ∈ ( 0 , 1 ) , are generators.

scholarly journals Faithful representations of infinite-dimensional nilpotent Lie algebras

2015 ◽  
Vol 27 (1) ◽  
Author(s):  
Ingrid Beltiţă ◽  
Daniel Beltiţă
Keyword(s):  
Banach Spaces ◽  
Lie Algebras ◽  
Locally Convex Space ◽  
Bounded Operators ◽  
Nilpotent Lie Algebras ◽  
Infinite Dimensional ◽  
Nilpotent Operators ◽  
Locally Convex ◽  
Special Case ◽  
Continuous Representations

AbstractFor locally convex, nilpotent Lie algebras we construct faithful representations by nilpotent operators on a suitable locally convex space. In the special case of nilpotent Banach–Lie algebras we get norm continuous representations by bounded operators on Banach spaces.

scholarly journals The Mackey convergence condition for spaces with webs

1988 ◽  
Vol 11 (3) ◽  
pp. 473-483 ◽  
Author(s):  
Thomas E. Gilsdorf
Keyword(s):  
Convex Space ◽  
Convergence Condition ◽  
Locally Convex Space ◽  
Locally Convex Spaces ◽  
Sufficient Condition ◽  
Locally Convex ◽  
Convex Spaces

If each sequence converging to0in a locally convex space is also Mackey convergent to0, that space is said to satisfy the Mackey convergence condition. The problem of characterizing those locally convex spaces with this property is still open. In this paper, spaces with compatible webs are used to construct both a necessary and a sufficient condition for a locally convex space to satisfy the Mackey convergence condition.

scholarly journals On the structure of support point sets

1990 ◽  
Vol 13 (1) ◽  
pp. 25-30
Author(s):  
E. Azoff ◽  
R. younis
Keyword(s):  
Convex Subset ◽  
Convex Space ◽  
Compact Convex ◽  
Extreme Points ◽  
Locally Convex Space ◽  
Support Point ◽  
Support Points ◽  
Point Set ◽  
Choquet's Theorem ◽  
Locally Convex

LetXbe a metrizable compact convex subset of a locally convex space. Using Choquet's Theorem, we determine the structure of the support point set ofXwhenXhas countably many extreme points. We also characterize the support points of certain families of analytic functions.

The spectra of Fredholm operators in locally convex spaces

1964 ◽  
Vol 60 (4) ◽  
pp. 801-806 ◽  
Author(s):  
P. A. Olagunju ◽  
T. T. West
Keyword(s):  
Sufficient Conditions ◽  
Trace Class ◽  
Locally Convex Space ◽  
Fredholm Operators ◽  
Negative Real Axis ◽  
Interesting Special Case ◽  
Necessary And Sufficient ◽  
Locally Convex ◽  
Special Case ◽  
General Banach Space

1. Notation and definitions. In this paper necessary and sufficient conditions are found for the spectrum of a Fredholm operator in a locally convex space (always taken to be Hausdorff) to lie on the non-negative real axis of the complex plane. Some results of Grothendieck(2) allow us to obtain the results in this general form; an interesting special case is the trace-class of operators in a general Banach space. We also deal with the case of Hilbert–Schmidt operators in a Hilbert space.

scholarly journals On the surjectivity of linear transformations

1996 ◽  
Vol 19 (3) ◽  
pp. 545-548
Author(s):  
M. Damlakhi ◽  
V. Anandam
Keyword(s):  
Reflexive Banach Space ◽  
Locally Convex Space ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Linear Transformations ◽  
Bounded Linear ◽  
Smooth Coefficients ◽  
Necessary And Sufficient ◽  
Locally Convex ◽  
Bounded Linear Transformation

LetBbe a reflexive Banach space,Xa locally convex space andT:B→X(not necessarily bounded) linear transformation. A necessary and sufficient condition is obtained so that for a givenv∈Xthere is a solution for the equationTu=v. This result is used to discuss the existence of anLp-weak solution ofDu=vwhereDis a differential operator with smooth coefficients andv∈Lp.

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