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.

An Embedding Theorem for Separable Locally Convex Spaces

1971 ◽  
Vol 14 (1) ◽  
pp. 119-120 ◽  
Author(s):  
Robert H. Lohman
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Convex Space ◽  
Separable Banach Space ◽  
Locally Convex Space ◽  
Embedding Theorem ◽  
Vector Spaces ◽  
Locally Convex Spaces ◽  
Topological Vector Spaces ◽  
Locally Convex

A well-known embedding theorem of Banach and Mazur [1, p. 185] states that every separable Banach space is isometrically isomorphic to a subspace of C[0, 1], establishing C[0, 1] as a universal separable Banach space. The embedding theorem one encounters in a course in topological vector spaces states that every Hausdorff locally convex space (l.c.s.) is topologically isomorphic to a subspace of a product of Banach spaces.

scholarly journals Injective absolutely summing operators

1988 ◽  
Vol 103 (3) ◽  
pp. 497-502
Author(s):  
Susumu Okada ◽  
Yoshiaki Okazaki
Keyword(s):  
Banach Space ◽  
Uniform Convergence ◽  
Convex Space ◽  
Locally Convex Space ◽  
Dimensional Banach Space ◽  
Infinite Dimensional ◽  
Infinite Dimensional Banach Space ◽  
Integrable Functions ◽  
Locally Convex ◽  
Convergence In Mean

Let X be an infinite-dimensional Banach space. It is well-known that the space of X-valued, Pettis integrable functions is not always complete with respect to the topology of convergence in mean, that is, the uniform convergence of indefinite integrals (see [14]). The Archimedes integral introduced in [9] does not suffer from this defect. For the Archimedes integral, functions to be integrated are allowed to take values in a locally convex space Y larger than the space X while X accommodates the values of indefinite integrals. Moreover, there exists a locally convex space Y, into which X is continuously embedded, such that the space ℒ(μX, Y) of Y-valued, Archimedes integrable functions is identical to the completion of the space of X valued, simple functions with repect to the toplogy of convergence in mean, for each non-negative measure μ (see [9]).

KAC-Moody Lie Algebras and the Classification of Nilpotent Lie Algebras of Maximal Rank

1982 ◽  
Vol 34 (6) ◽  
pp. 1215-1239 ◽  
Author(s):  
L. J. Santharoubane
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Finite Dimension ◽  
Maximal Rank ◽  
Killing Form ◽  
Nilpotent Lie Algebra ◽  
Nilpotent Lie Algebras ◽  
Infinite Dimensional ◽  
Dimensional Version

Introduction. The natural problem of determining all the Lie algebras of finite dimension was broken in two parts by Levi's theorem:1) the classification of semi-simple Lie algebras (achieved by Killing and Cartan around 1890)2) the classification of solvable Lie algebras (reduced to the classification of nilpotent Lie algebras by Malcev in 1945 (see [10])).The Killing form is identically equal to zero for a nilpotent Lie algebra but it is non-degenerate for a semi-simple Lie algebra. Therefore there was a huge gap between those two extreme cases. But this gap is only illusory because, as we will prove in this work, a large class of nilpotent Lie algebras is closely related to the Kac-Moody Lie algebras. These last algebras could be viewed as infinite dimensional version of the semisimple Lie algebras.

Necessary and Sufficient Conditions for the Equality of L(f) and l1

1982 ◽  
Vol 34 (2) ◽  
pp. 406-410 ◽  
Author(s):  
Waleed Deeb
Keyword(s):  
Banach Space ◽  
Convex Space ◽  
Sufficient Conditions ◽  
Dimensional Subspace ◽  
Locally Convex Space ◽  
Natural Question ◽  
Infinite Dimensional ◽  
Necessary And Sufficient ◽  
Locally Convex ◽  
Infinite Dimensional Subspace

Introduction. Let f be a modulus, ei = (δij) and E = {ei, i = 1, 2, …}. The L(f) spaces were created (to the best of our knowledge) by W. Ruckle in [2] in order to construct an example to answer a question of A. Wilansky. It turned out that these spaces are interesting spaces. For example lp, 0 < p ≦ 1 is an L(f) space with f(x) = xp, and every FK space contains an L(f) space [2]. A natural question is: For which f is L(f) a locally convex space? It is known that L(f) ⊆ l1, for all f modulus (see [2]), and l1 is the smallest locally convex FK space in which E is bounded (see [1]). Thus the question becomes: For which f does L(f) equal l1? In this paper we characterize such f. (An FK space need not be locally convex here.) We also characterize those f for which L(f) contains a convex ball. The final result of this paper is to show that if f satisfies f(x · y) ≦ f(x) · f(y) and L(f) ≠ l1 then L(f) contains no infinite dimensional subspace isomorphic to a Banach space.

scholarly journals On the closed graph theorem

1976 ◽  
Vol 17 (2) ◽  
pp. 89-97 ◽  
Author(s):  
J. O. Popoola ◽  
I. Tweddle
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Cardinal Number ◽  
Closed Graph ◽  
Graph Theorem ◽  
Closed Graph Theorem ◽  
Infinite Dimensional ◽  
Barrelled Spaces ◽  
Natural Sense ◽  
Locally Convex

Our main purpose is to describe those separated locally convex spaces which can serve as domain spaces for a closed graph theorem in which the range space is an arbitrary Banach space of (linear) dimension at most c, the cardinal number of the real line R. These are the δ-barrelled spaces which are considered in §4. Many of the standard elementary Banach spaces, including in particular all separable ones, have dimension at most c. Also it is known that an infinite dimensional Banach space has dimension at least c (see e.g. [8]). Thus if we classify Banach spaces by dimension we are dealing, in a natural sense, with the first class which contains infinite dimensional spaces.

scholarly journals Bounded generators in linear topological spaces

1971 ◽  
Vol 12 (2) ◽  
pp. 105-109
Author(s):  
S. O. Iyahen
Keyword(s):  
Banach Spaces ◽  
Convex Space ◽  
Locally Convex Space ◽  
Topological Spaces ◽  
Bounded Set ◽  
Definition Of ◽  
Bounded Sets ◽  
Locally Convex

Ito and Seidman in [5] define a BG space as a locally convex space in whichthere exists a bounded set with a dense span. In this note we extend the idea to a class of not necessarily locally convex linear topological spaces (l.t.s.). We note the link between the idea of a BG space and Weston’s characterization in [7] of separable Banach spaces. Finally we examine σ-BG spaces; here the bounded set in the definition of a BG space is replaced by the union of a sequence of bounded sets.

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 $$\omega ^\omega $$-Base and infinite-dimensional compact sets in locally convex spaces

2021 ◽  
Author(s):  
Taras Banakh ◽  
Jerzy Ka̧kol ◽  
Johannes Philipp Schürz
Keyword(s):  
Convex Space ◽  
Locally Convex Space ◽  
Compact Set ◽  
Compact Sets ◽  
Infinite Dimensional ◽  
Neighborhood Base ◽  
Remarkable Result ◽  
Locally Convex Topology ◽  
Locally Convex ◽  
Compact Subsets

AbstractA locally convex space (lcs) E is said to have an $$\omega ^{\omega }$$ ω ω -base if E has a neighborhood base $$\{U_{\alpha }:\alpha \in \omega ^\omega \}$$ { U α : α ∈ ω ω } at zero such that $$U_{\beta }\subseteq U_{\alpha }$$ U β ⊆ U α for all $$\alpha \le \beta $$ α ≤ β . The class of lcs with an $$\omega ^{\omega }$$ ω ω -base is large, among others contains all (LM)-spaces (hence (LF)-spaces), strong duals of distinguished Fréchet lcs (hence spaces of distributions $$D^{\prime }(\Omega )$$ D ′ ( Ω ) ). A remarkable result of Cascales-Orihuela states that every compact set in an lcs with an $$\omega ^{\omega }$$ ω ω -base is metrizable. Our main result shows that every uncountable-dimensional lcs with an $$\omega ^{\omega }$$ ω ω -base contains an infinite-dimensional metrizable compact subset. On the other hand, the countable-dimensional vector space $$\varphi $$ φ endowed with the finest locally convex topology has an $$\omega ^\omega $$ ω ω -base but contains no infinite-dimensional compact subsets. It turns out that $$\varphi $$ φ is a unique infinite-dimensional locally convex space which is a $$k_{\mathbb {R}}$$ k R -space containing no infinite-dimensional compact subsets. Applications to spaces $$C_{p}(X)$$ C p ( X ) are provided.

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 Stabilizing Solution for a Discrete-Time Modified Algebraic Riccati Equation in Infinite Dimensions

2015 ◽  
Vol 2015 ◽  
pp. 1-11 ◽  
Author(s):  
Viorica Mariela Ungureanu
Keyword(s):  
Banach Spaces ◽  
Discrete Time ◽  
Sufficient Conditions ◽  
Global Solutions ◽  
Trace Class ◽  
Algebraic Riccati Equation ◽  
Bounded Operators ◽  
Linear Quadratic ◽  
Infinite Dimensional ◽  
Ordered Banach Spaces

We provide necessary and sufficient conditions for the existence of stabilizing solutions for a class of modified algebraic discrete-time Riccati equations (MAREs) defined on ordered Banach spaces of sequences of linear and bounded operators. These MAREs arise in the study of linear quadratic (LQ) optimal control problems for infinite-dimensional discrete-time linear systems (DTLSs) affected simultaneously by multiplicative white noise (MN) and Markovian jumps (MJs). Unlike most of the previous works, where the detectability and observability notions are key tools for studying the global solvability of MAREs, in this paper the conditions of existence of mean-square stabilizing solutions are given directly in terms of system parameters. The methods we have used are based on the spectral theory of positive operators and the properties of trace class and compact operators. Our results generalise similar ones obtained for finite-dimensional MAREs associated with stochastic DTLSs without MJs. Also they complete and extend (in the autonomous case) former investigations concerning the existence of certain global solutions (as minimal, maximal, and stabilizing solutions) for generalized discrete-time Riccati type equations defined on infinite-dimensional ordered Banach spaces.

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