Closed graph theorems for generalized inductive limit topologies

1977 ◽  
Vol 82 (1) ◽  
pp. 67-83 ◽  
Author(s):  
W. Ruess
Keyword(s):  
Inductive Limit ◽  
Locally Convex Space ◽  
Subsequent Paper ◽  
Closed Graph ◽  
Strict Topology ◽  
Locally Compact ◽  
Special Cases ◽  
Bounded Sets ◽  
Locally Convex Topology ◽  
Locally Convex

SummaryThe object of this and a subsequent paper is to investigate the locally convex structure of several strict topologies that are generalizations of R. C. Buck's strict topology β on C(S), S locally compact Hausdorff. If the topology τ of a locally convex space (lcs) (X, τ) is any of these strict topologies, then it is localizable on every absorbing disc T in X, i.e. it is the finest locally convex topology on X agreeing with τ on T. Topologies of this kind are said to be (L)-topologies. As our main tools for the analysis of the structure of strict topologies, we deduce in this paper several closed graph theorems for spaces of type (L). In particular, it is shown that every semi-Montel lcs with a fundamental sequence of bounded sets and every Bτ-complete Schwartz space belongs to the class Bτ(L) of all lcs Y with the property that every closed linear map from any (L)-space X into Y is continuous. Further closed graph theorems are established and many of the known closed graph theorems are deduced as special cases of our results. Moreover, the problem of Bτ-completeness of locally convex spaces belonging to Bτ(L) is considered.

scholarly journals Strictly barrelled disks in inductive limits of quasi-(LB)-spaces

1996 ◽  
Vol 19 (4) ◽  
pp. 727-732
Author(s):  
Carlos Bosch ◽  
Thomas E. Gilsdorf
Keyword(s):  
Convex Space ◽  
Inductive Limit ◽  
Linear Span ◽  
Locally Convex Space ◽  
Minkowski Functional ◽  
Closed Graph ◽  
Inductive Limits ◽  
Locally Convex ◽  
Barrelled Space

A strictly barrelled diskBin a Hausdorff locally convex spaceEis a disk such that the linear span ofBwith the topology of the Minkowski functional ofBis a strictly barrelled space. Valdivia's closed graph theorems are used to show that closed strictly barrelled disk in a quasi-(LB)-space is bounded. It is shown that a locally strictly barrelled quasi-(LB)-space is locally complete. Also, we show that a regular inductive limit of quasi-(LB)-spaces is locally complete if and only if each closed bounded disk is a strictly barrelled disk in one of the constituents.

scholarly journals The strict topology on a space of vector-valued functions

1979 ◽  
Vol 22 (1) ◽  
pp. 35-41 ◽  
Author(s):  
Liaqat Ali Khan
Keyword(s):  
Scalar Field ◽  
Topological Space ◽  
Vector Space ◽  
Convex Space ◽  
Locally Convex Space ◽  
Strict Topology ◽  
Locally Compact ◽  
Vector Valued Functions ◽  
Vector Valued ◽  
Locally Convex

Let X be a topological space, E a real or complex topological vector space, and C(X, E) the vector space of all bounded continuous E-valued functions on X. The notion of the strict topology on C(X, E) was first introduced by Buck (1) in 1958 in the case of X locally compact and E a locally convex space. In recent years a large number of papers have appeared in the literature concerned with extending the results contained in Buck's paper (1); see, for example, (14), (15), (3), (4), (12), (2), and (6). Most of these investigations have been concerned with generalising the space X and taking E to be the scalar field or a locally convex space.

scholarly journals Stability and closed graph theorems in classes of bornological spaces

1982 ◽  
Vol 23 (2) ◽  
pp. 151-162
Author(s):  
T. K. Mukherjee ◽  
W. H. Summers
Keyword(s):  
Inductive Limit ◽  
Locally Convex Space ◽  
Vector Spaces ◽  
Locally Convex Spaces ◽  
Closed Graph ◽  
Quotient Spaces ◽  
Direct Sum ◽  
Bornological Space ◽  
Normed Vector ◽  
Locally Convex

In the general theory of locally convex spaces, the idea of inductive limit is pervasive, with quotient spaces and the less obvious notion of direct sum being among the instances. Bornological spaces provide another important example. As is well known (cf. [7]), a Hausdorff locally convex space E is bornological if, and only if, E is an inductive limit of normed vector spaces. Going even further in this direction, a complete Hausdorff bornological space is an inductive limit of Banach spaces.

Strict Topology on Paracompact Locally Compact Spaces

1977 ◽  
Vol 29 (1) ◽  
pp. 216-219 ◽  
Author(s):  
Surjit Singh Khurana
Keyword(s):  
Compact Space ◽  
Convex Space ◽  
Locally Convex Space ◽  
Complex Numbers ◽  
Strict Topology ◽  
Locally Compact ◽  
Compact Spaces ◽  
Locally Compact Space ◽  
Locally Compact Spaces ◽  
Locally Convex

In this paper, X denotes a Hausdorff paracompact locally compact space, E a Hausdorff locally convex space over K, the field of real or complex numbers (we call the elements of K scalars), a filtering upwards family of semi-norms on E generating the topology of E, Cb(X) the space of all continuous scalar-valued funcions on X, and Cb(X, E) the space of all continuous, bounded E-valued functions.

scholarly journals On the Generalized Weighted Lebesgue Spaces of Locally Compact Groups

2011 ◽  
Vol 2011 ◽  
pp. 1-15 ◽  
Author(s):  
I. Akbarbaglu ◽  
S. Maghsoudi
Keyword(s):  
Convex Space ◽  
Topological Algebra ◽  
Locally Convex Space ◽  
Locally Compact Group ◽  
Compact Groups ◽  
Topological Properties ◽  
Locally Compact ◽  
The Family ◽  
Locally Convex Topology ◽  
Locally Convex

Let be a locally compact group with a fixed left Haar measure and be a system of weights on . In this paper, we deal with locally convex space equipped with the locally convex topology generated by the family of norms . We study various algebraic and topological properties of the locally convex space . In particular, we characterize its dual space and show that it is a semireflexive space. Finally, we give some conditions under which with the convolution multiplication is a topological algebra and then characterize its closed ideals and its spectrum.

scholarly journals THE STRICT TOPOLOGY ON THE DISCRETE LEBESGUE SPACES

2010 ◽  
Vol 83 (2) ◽  
pp. 241-255 ◽  
Author(s):  
SAEID MAGHSOUDI ◽  
RASOUL NASR-ISFAHANI
Keyword(s):  
Positive Function ◽  
Locally Convex Space ◽  
Strict Topology ◽  
Convex Topology ◽  
Sigma 1 ◽  
Locally Convex Topologies ◽  
Second Dual ◽  
Locally Convex Topology ◽  
Locally Convex ◽  
Strong Dual

AbstractLet Σ be a set and σ be a positive function on Σ. We introduce and study a locally convex topology β1(Σ,σ) on the space ℓ1(Σ,σ) such that the strong dual of (ℓ1(Σ,σ),β1(Σ,σ)) can be identified with the Banach space $(c_0(\Sigma ,1/\sigma ),\|\cdot \|_{\infty ,\sigma })$. We also show that, except for the case where Σ is finite, there are infinitely many such locally convex topologies on ℓ1(Σ,σ). Finally, we investigate some other properties of the locally convex space (ℓ1(Σ,σ),β1(Σ,σ)) , and as an application, we answer partially a question raised by A. I. Singh [‘L∞0(G)* as the second dual of the group algebra L1 (G) with a locally convex topology’, Michigan Math. J.46 (1999), 143–150].

A CONVOLUTION-INDUCED TOPOLOGY ON THE ORLICZ SPACE OF A LOCALLY COMPACT GROUP

2015 ◽  
Vol 99 (1) ◽  
pp. 1-11
Author(s):  
IBRAHIM AKBARBAGLU ◽  
SAEID MAGHSOUDI
Keyword(s):  
Compact Group ◽  
Orlicz Space ◽  
Haar Measure ◽  
Locally Compact Group ◽  
Young Function ◽  
Strict Topology ◽  
Locally Compact ◽  
Induced Topology ◽  
Locally Convex Topology ◽  
Locally Convex

Let $G$ be a locally compact group with a fixed left Haar measure. In this paper, given a strictly positive Young function ${\rm\Phi}$, we consider $L^{{\rm\Phi}}(G)$ as a Banach left $L^{1}(G)$-module. Then we equip $L^{{\rm\Phi}}(G)$ with the strict topology induced by $L^{1}(G)$ in the sense of Sentilles and Taylor. Some properties of this locally convex topology and a comparison with weak$^{\ast }$, bounded weak$^{\ast }$ and norm topologies are presented.

scholarly journals STRICT TOPOLOGY AS A MIXED TOPOLOGY ON LEBESGUE SPACES

2011 ◽  
Vol 84 (3) ◽  
pp. 504-515 ◽  
Author(s):  
SAEID MAGHSOUDI ◽  
RASOUL NASR-ISFAHANI
Keyword(s):  
Topological Group ◽  
Compact Space ◽  
Group Algebra ◽  
Lebesgue Space ◽  
Strict Topology ◽  
Lebesgue Spaces ◽  
Locally Compact ◽  
Locally Convex Topology ◽  
Locally Convex ◽  
Strong Dual

AbstractLetXbe a locally compact space, and 𝔏∞0(X,ι) be the space of all essentially boundedι-measurable functionsfonXvanishing at infinity. We introduce and study a locally convex topologyβ1(X,ι) on the Lebesgue space 𝔏1(X,ι) such that the strong dual of (𝔏1(X,ι),β1(X,ι)) can be identified with$({\frak L}_0^\infty (X,\iota ),\|\cdot \|_\infty )$. Next, by showing thatβ1(X,ι) can be considered as a natural mixed topology, we deduce some of its basic properties. Finally, as an application, we prove thatL1(G) , the group algebra of a locally compact Hausdorff topological groupG, equipped with the convolution multiplication is a complete semitopological algebra under theβ1(G) topology.

scholarly journals Continuity of Convolution of Test Functions on Lie Groups

2014 ◽  
Vol 66 (1) ◽  
pp. 102-140
Author(s):  
Lidia Birth ◽  
Helge Glöckner
Keyword(s):  
Continuous Functions ◽  
Locally Convex Space ◽  
Locally Compact Group ◽  
Locally Convex Spaces ◽  
Test Functions ◽  
Bilinear Map ◽  
Locally Compact ◽  
Compactly Supported ◽  
Locally Convex

AbstractFor a Lie group G, we show that the map taking a pair of test functions to their convolution, is continuous if and only if G is σ-compact. More generally, consider with t ≤ r + s, locally convex spaces E1, E2 and a continuous bilinear map b : E1 × E2 → F to a complete locally convex space F. Let be the associated convolution map. The main result is a characterization of those (G; r; s; t; b) for which β is continuous. Convolution of compactly supported continuous functions on a locally compact group is also discussed as well as convolution of compactly supported L1-functions and convolution of compactly supported Radon measures.

scholarly journals A generalisation of Mackey's theorem and the uniform boundedness principle

1989 ◽  
Vol 40 (1) ◽  
pp. 123-128 ◽  
Author(s):  
Charles Swartz
Keyword(s):  
Uniform Boundedness ◽  
Mackey Topology ◽  
Convex Topology ◽  
Uniform Boundedness Principle ◽  
Bounded Sets ◽  
Locally Convex Topology ◽  
Locally Convex

We construct a locally convex topology which is stronger than the Mackey topology but still has the same bounded sets as the Mackey topology. We use this topology to give a locally convex version of the Uniform Bouudedness Principle which is valid without any completeness or barrelledness assumptions.

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