Topological Vector Spaces of Continuous Functions

Nachbin, Léopoldo

doi:10.1073/pnas.40.6.471

Cited by 87 publications

(33 citation statements)

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“…As an impovement on this result we have the following proposition. The proof is a slight variation of the proof of a result of Nachbin's ( [10], Th. 1) and is therefore omitted.…”

Section: Where Hec(k) and H' Is Any Continuous Extension Of H To All mentioning

confidence: 99%

“…According to a result of Nachbin [10] and Shirota [12], C(X), for X completely regular Hausdorff, is bornological if, and only if, X is realcompact. From the fact that a complete bornological space is barrelled, we have: COROLLARY …”

Section: Let E Be a Complete *-Algebra Then E Is Topologically *-Isomentioning

confidence: 99%

“…Note that, by (6.1), a closed C(M)-pseudocompact subset of M is compact. According to a well known result due independently to Nachbin [10] and Shirota [12], the real space C(X, R) is barrelled if, and only if, every C(X, ϋ? )-pseudocompact subset of X is compact.…”

Section: Let X Be a Realcompact Space And T A Topology On C(x) Such Tmentioning

confidence: 99%

“…For example, letting R denote the real numbers, C(R) is barrelled and therefore has the Mackey topology (see [10] or [12]). The theorem has some point, however, since not every C(X) has the Mackey topology.…”

Section: Where Hec(k) and H' Is Any Continuous Extension Of H To All mentioning

confidence: 99%

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Functional representation of topological algebras

Morris¹,

Wulbert²

1967

Pacific J. Math.

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“…As an impovement on this result we have the following proposition. The proof is a slight variation of the proof of a result of Nachbin's ( [10], Th. 1) and is therefore omitted.…”

Section: Where Hec(k) and H' Is Any Continuous Extension Of H To All mentioning

confidence: 99%

Section: Let E Be a Complete *-Algebra Then E Is Topologically *-Isomentioning

confidence: 99%

Section: Let X Be a Realcompact Space And T A Topology On C(x) Such Tmentioning

confidence: 99%

Section: Where Hec(k) and H' Is Any Continuous Extension Of H To All mentioning

confidence: 99%

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Functional representation of topological algebras

Morris¹,

Wulbert²

1967

Pacific J. Math.

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“…[7,13]). In the more general setting of weighted spaces, Ernst and Schnettler [3] studied the (gDF) and the quasi-barrelledness of weighted spaces by constructing a Nachbin family on X which is based on the assumption that the weighted space has a fundamental sequence of bounded sets.…”

Section: Introduction and Notationsmentioning

confidence: 99%

On weighted spaces without a fundamental sequence of bounded sets

Olaleru

2002

International Journal of Mathematics and Mathematical Sciences

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The problem of countably quasi-barrelledness of weighted spaces of continuous functions, of which there are no results in the general setting of weighted spaces, is tackled in this paper. This leads to the study of quasi-barrelledness of weighted spaces in which, unlike that of Ernst and Schnettler (1986), though with a similar approach, we drop the assumption that the weighted space has a fundamental sequence of bounded sets. The study of countably quasi-barrelledness of weighted spaces naturally leads to definite results on the weighted (DF)-spaces for those weighted spaces with a fundamental sequence of bounded sets.2000 Mathematics Subject Classification: 46A08, 46E30. Introduction and notations.The countably barrelledness, countably quasi-barrelledness, barrelledness, and quasi-barrelleness of the space C(X) of continuous functions on a completely regular Hausdorff space X equipped with the compact open topology (c-op), including when it is a (DF)-space and (gDF)-space, is well known (cf. [7,13]). In the more general setting of weighted spaces, Ernst and Schnettler [3] studied the (gDF) and the quasi-barrelledness of weighted spaces by constructing a Nachbin family on X which is based on the assumption that the weighted space has a fundamental sequence of bounded sets. A weighted space need not have a fundamental system of bounded sets. There are classical examples of such spaces, for example, let X be a noncompact locally compact and σ -compact space, and let C(X) be the space of all continuous real-valued functions on X equipped with the compact-open topology. C(X) is a weighted space [14], metrizable and not normable [11, Observation 10.1.25] and hence it does not have a fundamental sequence of bounded sets [6]. See also [10, page 9] for another example. Following the same approach, the countably barrelledness and barrelledness of weighted spaces was studied in [10] without assuming that the weighted space has a fundamental sequence of bounded sets. This paper is a follow up of [10] although we have made it independent of it. We characterise countably quasi-barrelled (Section 3) and quasi-barrelled (Section 4) weighted spaces by a constructed Nachbin family on X without assuming that the spaces have a fundamental sequence of bounded sets. This approach makes the study of (DF)-weighted spaces easy as it can be seen in Section 5. In Section 6, we show that every countably quasi-barrelled weighted space satisfies the countable neighborhood property (cnp).As an application, we recover the known results for the countably quasi-barrelledness and quasi-barrelledness of C(X) equipped with compact open topology (Corollary 3.10) and at the same time get new results for the space of bounded continuous

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Über einige Limitierungen auf 𝒞(X) und den Satz von Dini

Kutzler¹

1974

Mathematische Nachrichten

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Eiiileitung und BezeichnungsweisenDer Satz von DINI ist in der Theorie der reellwertigen, stetigen Funktionen auf einem kompakten bzw. lokalkompaliten topologischen Rauin ein wichtiges Hilfsmittel, um aus der punktweisen Konvergenz einer gerichteteii Faniilie stetiger Funktionen gegen eine stetige Funktion auf die gleichmaoige Konvergenz hzw. Konvergenz in der kompakt-offenen Topologie zu schliel3en. I n diesem Zusainmenhang hat der Satz von DINI mehr technischen Charakter. Die folgenden Untersuchungen sollen nun zeigen, daB dieser Satz irn Kahinen des Studiuiris to1)ologischer Raume mit funktionalanalytischen Methoden auch zu strukturellen Aussagen fahig ist.Es ist wohlbekannt, daIJ fur einen lokalkompakten, topologischen HAUSDORFFrzturn T die kompakt-offene Topologie auf d ( T ) == G(T, R ) universe11 charakterisierbar ist als die grobste Toplogie auf iY(T) mit der Eigenschaft, daB die Evaluationsabbildung ev : b (T) >: T -+ €2, definiert durch e r i ( f , p ) = f ( p ) fur (f, p ) aus iY( T) x T, stetig ist. Ferner ist bekannt, dal3 sich auf d (T) keine T o plogie rnit der soehen erwahnten Eigensc~haft finden lafit, falls T vollstandig reyular aher nicht lokalkompakt ist. l m Rahnien der Theorie der Liniesraunie, die u. a. voii H. R. FTSCHER 171 begruiidet wurde, definierten RINZ und KELLER [3] sowie COOK und FISCHER [ 5 ] wie auch andere ilutoren den Begriff der stetigen Konvergenz, der das oben aufgefuhrte, universelle Prohlem im Rahmen der Theorie cler Lirnesriiuine loste. Fiir einen Limesraum X sol1 im Folgenden d ( . ( X ) stets die niit der Limitierung der stetigeii Konvergenz versehene Algebra iY ( X ) aller stet igen, reellwertigen, auf S definierten Funktionen bezeichnen.

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Topological Vector Spaces of Continuous Functions

Cited by 87 publications

References 1 publication

Functional representation of topological algebras

Functional representation of topological algebras

On weighted spaces without a fundamental sequence of bounded sets

Über einige Limitierungen auf 𝒞(X) und den Satz von Dini

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