Lines and Hyperplanes associated with Families of Closed and Bounded Sets in Conjugate Banach Spaces

1970 ◽  
Vol 22 (5) ◽  
pp. 933-938
Author(s):  
M. Edelstein
Keyword(s):  
General Setting ◽  
Locally Convex Space ◽  
Finite Family ◽  
Topological Spaces ◽  
Compact Sets ◽  
Straight Line ◽  
Straight Lines ◽  
Bounded Sets ◽  
Infinite Set ◽  
Locally Convex

Let be a family of sets in a linear space X. A hyperplane π is called a k-secant of if π intersects exactly k members of . The existence of k-secants for families of compact sets in linear topological spaces has been discussed in a number of recent papers (cf. [3–7]). For X normed (and a finite family of two or more disjoint non-empty compact sets) it was proved [5] that if the union of all members of is an infinite set which is not contained in any straight line of X, then has a 2-secant. This result and related ones concerning intersections of members of by straight lines have since been extended in [4] to the more general setting of a Hausdorff locally convex space.

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 Almost-periodicity in linear topological spaces and applications to abstract differential equations

1984 ◽  
Vol 7 (3) ◽  
pp. 529-540 ◽  
Author(s):  
Gaston Mandata N'Guerekata
Keyword(s):  
Continuous Function ◽  
Real Line ◽  
Locally Convex Space ◽  
Topological Spaces ◽  
Almost Periodicity ◽  
Almost Periodic ◽  
Fréchet Spaces ◽  
Abstract Differential Equations ◽  
T Tau ◽  
Locally Convex

LetEbe a complete locally convex space (l.c.s.) andf:R→Ea continuous function; thenfis said to be almost-periodic (a.p.) if, for every neighbourhood (of the origin inE)U, there existsℓ=ℓ(U)>0such that every interval[a,a+ℓ]of the real line contains at least oneτpoint such thatf(t+τ)−f(t)∈Ufor everyt∈R. We prove in this paper many useful properties of a.p. functions in l.c.s, and give Bochner's criteria in Fréchet spaces.

scholarly journals NOTE ABOUT LINDELÖF Σ-SPACES υX

2011 ◽  
Vol 85 (1) ◽  
pp. 114-120
Author(s):  
J. KA̧KOL ◽  
M. LÓPEZ-PELLICER
Keyword(s):  
Functional Analysis ◽  
Convex Space ◽  
Locally Convex Space ◽  
Countable Tightness ◽  
Pointwise Topology ◽  
Bounded Sets ◽  
General Fact ◽  
Locally Convex

AbstractThe paper deals with the following problem: characterize Tichonov spaces X whose realcompactification υX is a Lindelöf Σ-space. There are many situations (both in topology and functional analysis) where Lindelöf Σ (even K-analytic) spaces υX appear. For example, if E is a locally convex space in the class 𝔊 in sense of Cascales and Orihuela (𝔊 includes among others (LM ) -spaces and (DF ) -spaces), then υ(E′,σ(E′,E)) is K-analytic and E is web-bounded. This provides a general fact (due to Cascales–Kakol–Saxon): if E∈𝔊, then σ(E′,E) is K-analytic if and only if σ(E′,E) is Lindelöf. We prove a corresponding result for spaces Cp (X) of continuous real-valued maps on X endowed with the pointwise topology: υX is a Lindelöf Σ-space if and only if X is strongly web-bounding if and only if Cp (X) is web-bounded. Hence the weak* dual of Cp (X) is a Lindelöf Σ-space if and only if Cp (X) is web-bounded and has countable tightness. Applications are provided. For example, every E∈𝔊 is covered by a family {Aα :α∈Ω} of bounded sets for some nonempty set Ω⊂ℕℕ.

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.

(HM)-Spaces and Measurable Cardinals

1984 ◽  
Vol 27 (1) ◽  
pp. 53-57
Author(s):  
José A. Facenda Aguirre
Keyword(s):  
Open Problem ◽  
Nonstandard Analysis ◽  
Measurable Cardinal ◽  
Locally Convex Space ◽  
Continuous Norm ◽  
Finite Dimensional ◽  
Reflexive Space ◽  
Measurable Cardinals ◽  
Bounded Sets ◽  
Locally Convex

AbstractA locally convex space E is called an (HM)-space if E has invariant nonstandard hulls. In this paper we prove that if E is an (HM)-space, then E is a T(μ)-space, where μ is the first measurable cardinal. This is equivalent to say that in an (HM)-space, with dim(E)≧μ, does not exist a continuous norm. With this result, we prove that there exists an inductive semi-reflexive space E such that the bounded sets in E are finite-dimensional but E is not an (HM)-space. Thus, we answer negatively to an open problem raised up by Bellenot. In this paper, we do not use nonstandard analysis.

scholarly journals Unconditional convergence and bases

1973 ◽  
Vol 18 (4) ◽  
pp. 321-324 ◽  
Author(s):  
I. Tweddle
Keyword(s):  
Convex Space ◽  
Present Note ◽  
General Setting ◽  
Locally Convex Space ◽  
Unconditional Convergence ◽  
Weak Basis ◽  
Basis Theorem ◽  
Locally Convex ◽  
Extended Basis

The recent papers (6), (7) of J. T. Marti have revived interest in the concept of extended bases, introduced in (1) by M. G. Arsove and R. E. Edwards. In the present note, two results are established which involve this idea. The first of these, which is given in a more general setting, restricts the behaviour of the coefficients for an extended basis in a certain type of locally convex space. The second result extends the well-known weak basis theorem (1, Theorem 11).

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.

Automatic Tread Gaging Machine

1979 ◽  
Vol 7 (1) ◽  
pp. 31-39
Author(s):  
G. S. Ludwig ◽  
F. C. Brenner
Keyword(s):  
Experimental Tests ◽  
Automatic Machine ◽  
Wear Rates ◽  
Handling Device ◽  
Straight Line ◽  
Component Systems ◽  
Before And After ◽  
Good Repeatability ◽  
The Difference ◽  
Straight Lines

Abstract An automatic tread gaging machine has been developed. It consists of three component systems: (1) a laser gaging head, (2) a tire handling device, and (3) a computer that controls the movement of the tire handling machine, processes the data, and computes the least-squares straight line from which a wear rate may be estimated. Experimental tests show that the machine has good repeatability. In comparisons with measurements obtained by a hand gage, the automatic machine gives smaller average groove depths. The difference before and after a period of wear for both methods of measurement are the same. Wear rates estimated from the slopes of straight lines fitted to both sets of data are not significantly different.

Mackey Continuity of Characteristic Functionals

2002 ◽  
Vol 9 (1) ◽  
pp. 83-112
Author(s):  
S. Kwapień ◽  
V. Tarieladze
Keyword(s):  
Product Space ◽  
Locally Convex Space ◽  
Inner Product ◽  
Nuclear Space ◽  
Probability Measures ◽  
Linear Kernel ◽  
Characteristic Functional ◽  
Initial Space ◽  
Radon Probability Measure ◽  
Locally Convex

Abstract Problems of the Mackey-continuity of characteristic functionals and the localization of linear kernels of Radon probability measures in locally convex spaces are investigated. First the class of spaces is described, for which the continuity takes place. Then it is shown that in a non-complete sigmacompact inner product space, as well as in a non-complete sigma-compact metizable nuclear space, there may exist a Radon probability measure having a non-continuous characteristic functional in the Mackey topology and a linear kernel not contained in the initial space. Similar problems for moment forms and higher order kernels are also touched upon. Finally, a new proof of the result due to Chr. Borell is given, which asserts that any Gaussian Radon measure on an arbitrary Hausdorff locally convex space has the Mackey-continuous characteristic functional.

scholarly journals A maximum principle

1979 ◽  
Vol 28 (1) ◽  
pp. 23-26
Author(s):  
Kung-Fu Ng
Keyword(s):  
Maximum Principle ◽  
Extreme Point ◽  
Convex Space ◽  
Maximal Element ◽  
Locally Convex Space ◽  
Compact Set ◽  
Natural Manner ◽  
Upper Semicontinuous ◽  
Nonempty Compact ◽  
Locally Convex

AbstractLet K be a nonempty compact set in a Hausdorff locally convex space, and F a nonempty family of upper semicontinuous convex-like functions from K into [–∞, ∞). K is partially ordered by F in a natural manner. It is shown among other things that each isotone, upper semicontinuous and convex-like function g: K → [ – ∞, ∞) attains its K-maximum at some extreme point of K which is also a maximal element of K.Subject classification (Amer. Math. Soc. (MOS) 1970): primary 46 A 40.

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