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.

On the fixed point index in locally convex spaces

1987 ◽  
Vol 106 (1-2) ◽  
pp. 161-168 ◽  
Author(s):  
M. Furi ◽  
M. P. Pera
Keyword(s):  
Fixed Point Index ◽  
Eigenvalue Problems ◽  
Index Theory ◽  
Locally Convex Space ◽  
Locally Convex Spaces ◽  
Point Index ◽  
Locally Compact ◽  
Nonlinear Eigenvalue Problems ◽  
Continuation Principle ◽  
Locally Convex

SynopsisLet E be a Hausdorff locally convex space, Q a convex closed subset of E and U an open subset of Q. We develop an index theory for a class of locally compact maps f: U → E for which the usual assumption f(U) ⊂ Q is replaced by an appropriate “pushing condition”. Moreover, from this index theory, we deduce a general continuation principle and some global results for nonlinear eigenvalue problems.

scholarly journals Multipliers from spaces of test functions to amalgams

1993 ◽  
Vol 54 (1) ◽  
pp. 97-110
Author(s):  
Maria Torres De Squire
Keyword(s):  
Fourier Transform ◽  
Abelian Group ◽  
Compact Abelian Group ◽  
Continuous Functions ◽  
Locally Compact Abelian Group ◽  
Test Functions ◽  
Space Of Continuous Functions ◽  
Locally Compact ◽  
The Fourier Transform

AbstractIn this paper we study the space of multipliers M(r, s: p, q) from the space of test functions Φrs(G), on a locally compact abelian group G, to amalgams (Lp, lq)(G); the former includes (when r = s = ∞) the space of continuous functions with compact support and the latter are extensions of the Lp(G) spaces. We prove that the space M(∞: p) is equal to the derived space (Lp)0 defined by Figá-Talamanca and give a characterization of the Fourier transform for amalgams in terms of these spaces of multipliers.

On ∧-Mackey convergence in locally convex spaces

1984 ◽  
Vol 96 (3) ◽  
pp. 495-500
Author(s):  
Jan H. Fourie
Keyword(s):  
Convergence Property ◽  
Locally Convex Space ◽  
Locally Convex Spaces ◽  
Necessary And Sufficient Condition ◽  
Banach Sequence Space ◽  
Necessary And Sufficient ◽  
Locally Convex ◽  
Banach Sequence ◽  
Convex Spaces

In this note we introduce the concepts of Λ-Mackey sequence, Λ-Mackey convergence property, Λ-Schwartz family and associated Λ-Schwartz family and consider some applications of these to locally convex spaces. Hereby Λ denotes a Banach sequence space with the AK-property — the results of this paper generalize those in [4] where the case Λ = I1 is considered. We obtain a dual characterization of those locally convex spaces which satisfy the Λ-Mackey convergence property and characterize the dual Λ-Schwartz spaces in terms of the SM-property which is introduced in [10]. Finally, necessary and sufficient condition for a locally convex space to be ultra-bornological is proved.

scholarly journals Some Intersections of the Weighted -Spaces

2013 ◽  
Vol 2013 ◽  
pp. 1-12 ◽  
Author(s):  
F. Abtahi ◽  
H. G. Amini ◽  
H. A. Lotfi ◽  
A. Rejali
Keyword(s):  
Banach Space ◽  
Sufficient Conditions ◽  
Locally Convex Space ◽  
Locally Compact Group ◽  
Weight Functions ◽  
Convolution Product ◽  
Arbitrary Subset ◽  
Locally Compact ◽  
Algebraic Properties ◽  
Locally Convex

Let be a locally compact group an arbitrary family of the weight functions on and . The locally convex space as a subspace of is defined. Also, some sufficient conditions for that space to be a Banach space are provided. Furthermore, for an arbitrary subset of and a positive submultiplicative weight function on , Banach subspace of is introduced. Then some algebraic properties of , as a Banach algebra under convolution product, are investigated.

scholarly journals THE STRONG DUAL OF MEASURE ALGEBRAS WITH CERTAIN LOCALLY CONVEX TOPOLOGIES

2013 ◽  
Vol 87 (3) ◽  
pp. 353-365 ◽  
Author(s):  
HOSSEIN JAVANSHIRI ◽  
RASOUL NASR-ISFAHANI
Keyword(s):  
Banach Space ◽  
Convex Space ◽  
Closed Subspace ◽  
Locally Convex Space ◽  
Locally Compact Group ◽  
Measure Algebra ◽  
Locally Compact ◽  
Locally Convex Topologies ◽  
Locally Convex ◽  
Strong Dual

AbstractFor a locally compact group $ \mathcal{G} $, we introduce and study a class of locally convex topologies $\tau $ on the measure algebra $M( \mathcal{G} )$ of $ \mathcal{G} $. In particular, we show that the strong dual of $(M( \mathcal{G} ), \tau )$ can be identified with a closed subspace of the Banach space $M\mathop{( \mathcal{G} )}\nolimits ^{\ast } $; we also investigate some properties of the locally convex space $(M( \mathcal{G} ), \tau )$.

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 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.

Some criteria for nuclearity

1986 ◽  
Vol 100 (1) ◽  
pp. 151-159 ◽  
Author(s):  
M. A. Sofi
Keyword(s):  
Sequence Space ◽  
Convex Space ◽  
Locally Convex Space ◽  
Sequence Spaces ◽  
Bilinear Forms ◽  
Simple Formulation ◽  
Normal Topology ◽  
Nuclear Spaces ◽  
Locally Convex

For a given locally convex space, it is always of interest to find conditions for its nuclearity. Well known results of this kind – by now already familiar – involve the use of tensor products, diametral dimension, bilinear forms, generalized sequence spaces and a host of other devices for the characterization of nuclear spaces (see [9]). However, it turns out, these nuclearity criteria are amenable to a particularly simple formulation in the setting of certain sequence spaces; an elegant example is provided by the so-called Grothendieck–Pietsch (GP, for short) criterion for nuclearity of a sequence space (in its normal topology) in terms of the summability of certain numerical sequences.

scholarly journals A characterization of Pontryagin–van Kampen duality for locally convex spaces

2002 ◽  
Vol 121 (1-2) ◽  
pp. 75-89 ◽  
Author(s):  
Fernando Garibay Bonales ◽  
F.Javier Trigos-Arrieta ◽  
Rigoberto Vera Mendoza
Keyword(s):  
Locally Convex Spaces ◽  
Locally Convex ◽  
Convex Spaces

scholarly journals On some optimization problems in not necessarily locally convex space

2008 ◽  
Vol 18 (2) ◽  
pp. 167-172
Author(s):  
Ljiljana Gajic
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Existence Theorem ◽  
Convex Space ◽  
Optimization Problems ◽  
Locally Convex Space ◽  
Locally Convex Spaces ◽  
Equilibrium Existence ◽  
Cooperative Equilibrium ◽  
Locally Convex

In this note, by using O. Hadzic's generalization of a fixed point theorem of Himmelberg, we prove a non - cooperative equilibrium existence theorem in non - compact settings and a generalization of an existence theorem for non - compact infinite optimization problems, all in not necessarily locally convex spaces.

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