Spaces of vector-valued continuous functions

Schmets, Jean

doi:10.1007/bfb0066857

Cited by 12 publications

(20 citation statements)

References 6 publications

(3 reference statements)

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“…[4]). If (L, 1111) = co the positive answer is a particular case of a result of Mujica [5,1,7] . We preve now that the answer is positive for arbitrary (L, 1) 11) satisfying (E) .…”

Section: N-oomentioning

confidence: 99%

“…The equicontinous sequence (un)nEN defines a continous seminorm as follows : P(X) = sup {¡un(x)1 ; n E N} Thus (p(xk))kEN E L, a contradiction, since yk < P(xk) for all k E NI . 5. Remark: For an inductive limit E = ind E n and a normal Banach sequence space (L, 1111), the algebraic coincidente L(E) = ind L(En ) is a clearly equivalent to dx E L(E) In E IN with x E L(E n ) .…”

Section: N-oomentioning

confidence: 99%

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Inductive limits of vector-valued sequence spaces

Bonet¹,

Dierolf²,

Fernández³

1989

Publicacions Matemàtiques

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“…[4]). If (L, 1111) = co the positive answer is a particular case of a result of Mujica [5,1,7] . We preve now that the answer is positive for arbitrary (L, 1) 11) satisfying (E) .…”

Section: N-oomentioning

confidence: 99%

Section: N-oomentioning

confidence: 99%

Inductive limits of vector-valued sequence spaces

Bonet¹,

Dierolf²,

Fernández³

1989

Publicacions Matemàtiques

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“…A survey of what is known about spaces of real valued or real vector space valued continuous functions can be found in [9] and [11]. REMARK 2.3.…”

Section: 2mentioning

confidence: 99%

Duality properties of spaces of non-Archimedean valued functions

Govaerts

1987

J Aust Math Soc A

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Let C(X, F) be the space of all continuous functions from the ultraregular compact Hausdorff space X into the separated locally Jf-convex space F; AT is a complete, but not necessarily spherically complete, non-Archimedean valued field and C(X, F) is provided with the topology of uniform convergence on X. We prove that C( X, F) is ^-barrelled (respectively AT-quasibarrelled) if and only if F is AT-barrelled (respectively /f-quasibarrelled). This is not true in the case of R or C-valued functions. No complete characterization of the if-bornological spaces C( X, F) is obtained, but our results are, nevertheless, slightly better than the Archimedean ones. Finally, we introduce a notion of A"-ultrabornological spaces for K non-spherically complete and use it to study Af-ultrabornological spaces C(X, F).1980 Mathematics subject classification (Amer. Math. Soc.): 46 P 05.

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“…Conversely, let (g' n ) be an equicontinuous and σ(V(X, E)\ V(X 9 £))-null sequence. By [14,III.3 and III.4], there exist a compact subset K of X and an equicontinuous sequence…”

Section: 3])mentioning

confidence: 99%

“…By [14,IΠ.4.5] there exists a continuous seminorm p in E such that V p μ n (X) < 1, for every n e N, where μ n is the representing measure of g f n [14,III]. Let T = Σ n 2~nV p μ n .…”

Section: Theorem Let X Be a Completely Regular Hausdorff G-space Andmentioning

confidence: 99%

Grothendieck locally convex spaces of continuous vector valued functions

Freniche¹

1985

Pacific J. Math.

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Let ^{X, E) be the space of continuous functions from the completely regular Hausdorff space X into the Hausdorff locally convex space E, endowed with the compact-open topology. Our aim is to characterize the ^(X, E) spaces which have the following property: weak-star and weak sequential convergences coincide in the equicontinuous subsets of ^(X, E)'. These spaces are here called Grothendieck spaces. It is shown that in the equicontinuous subsets of E' the σ(E', E)-and β(E', ^-sequential convergences coincide, if ^(X, E) is a Grothendieck space and X contains an infinite compact subset. Conversely, if X is a G-space and E is a strict inductive limit of Frechet-Montel spaces ^(X, E) is a Grothendieck space. Therefore, it is proved that if £ is a separable Frechet space, then E is a Montel space if and only if there is an infinite compact Hausdorff X such that , E) is a Grothendieck space.1. Introduction. In this paper X will always denote a completely regular Hausdorff topological space, E a Hausdorff locally convex space, and &( X, E) the space of continuous functions from Xinto E, endowed with the compact-open topology. When E is the scalar field of reals or complex numbers, we write ^(X) instead ^( X, E).It is well known that Ή{X, E) is a Montel space whenever ^(X) and E so are, hence, if and only if X is discrete and E is a Montel space (see [5] , [16]).We study what happens when X has the following weaker property: the compact subsets of X are G-spaces (see below for definitions).We obtain in Theorem 4.4 that if £ is a Frechet-Montel space and X has that property, then ^( X, E) is a Grothendieck locally convex space. The key in the proof is the following fact: every countable equicontinuous subset of ^(X, E)' lies, via a Radon-Nikodym theorem, in a suitable L\τ 9 Eβ). As a consequence of a theorem of Mύjica [10], the same result is true when E is a strict inductive limit of Frechet-Montel spaces.In §3 we study the converse of 4.4. In Corollary 3.3 it is proved that if X contains an infinite compact subset, E is a Frechet separable space and #( X, E) is a Grothendieck space, then E is a Montel space. This property characterizes the Montel spaces among the Frechet separable spaces.

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Spaces of vector-valued continuous functions

Abstract: In: Josef Novák (ed.): General topology and its relations to modern analysis and algebra IV,

Cited by 12 publications

References 6 publications

Inductive limits of vector-valued sequence spaces

Inductive limits of vector-valued sequence spaces

Duality properties of spaces of non-Archimedean valued functions

Grothendieck locally convex spaces of continuous vector valued functions

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