On bounded sets of holomorphic mappings

Barroso, Jorge Alberto; Matos, Mário C.; Nachbin, Léopoldo

doi:10.1007/bfb0069011

Cited by 15 publications

(5 citation statements)

References 4 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…But if U<zE where E is a &JKyV~ space or a Prechet space, then τ 0 bounded subsets of H(U) are locally uniformly bounded [1]. Hence (f a°Πn ) aeA is locally uniformly bounded and therefore so is \Ja)aeA'…”

Section: /(«• + »•) -/Go ^ O•mentioning

confidence: 99%

Holomorphy on spaces of distribution

Boland¹,

Dineen²

1981

Pacific J. Math.

View full text Add to dashboard Cite

Section: /(«• + »•) -/Go ^ O•mentioning

confidence: 99%

Holomorphy on spaces of distribution

Boland¹,

Dineen²

1981

Pacific J. Math.

View full text Add to dashboard Cite

“…We recall the definition of a holomorphically barrelled space (see [4] and [3] for the motivation and some properties of this kind of space). In order to prove this theorem we need a lemma.…”

Section: (B) the Mapping F Is S-holomorphic On Umentioning

confidence: 99%

“…Thus E B is a Banach space since E is quasi-complete. Now E B is holomorphically barrelled (see [4] and [3]). Since SC \ U n £ B is T 0/ -bounded in U nE B for the normed topology, it is amply bounded in U r\E B for the normed topology.…”

Section: (B) the Mapping F Is S-holomorphic On Umentioning

confidence: 99%

“…It is not possible to erase the condition "ExF holomorphically bornological" in Corollary 3.6. In [4] and [7] the authors show that C N x C m is not holomorphically infrabarrelled, hence it is not holomorphically bornological, although C^ is a Frechet space and C (N) is a Silva space. Now we show that the Corollary 3.6 does not hold true in the case E = C™ and F = C (N) .…”

Section: Corollary Let E and F Be Quasi-complete Holomorphically Bormentioning

confidence: 99%

See 1 more Smart Citation

Holomorphically Bornological Spaces and Infinite-Dimensional Versions of Hartogs' Theorem

Matos

1978

Journal of the London Mathematical Society

Self Cite

View full text Add to dashboard Cite

A notion of holomorphically bornological space is introduced and several results are proved for spaces of this type. Proofs of infinite-dimensional versions of the Hartogs' Theorem on separately holomorphic mappings are given. NotationIn this paper unless otherwise stated E and F are complex locally convex spaces and U is a non-void open subset of E. Let @ E denote the family of all closed absolutely convex bounded subsets of E. lfBe& E , let E B be the vector subspace of £ generated by B and considered with the normed topology given by the Minkowsky functional determined by B. Let 2f? (V; F) indicate the vector space of all holomorphic mappings from U into F. For basic properties of this space see [12], [13] and [2].

show abstract

“…In Barroso, Matos and Nachbin [1] it has been shown that when E is a Silva space, all of the standard topologies Ts,tu,t0 (= e) coincide on H(E). Hence in particular we know that if F is a VFN space, then all of the standard topologies t6, tw, r0 on H(E) are equivalent Fréchet nuclear topologies.…”

Section: Topologies On H(e)mentioning

confidence: 99%

Holomorphic Functions on Nuclear Spaces

Boland¹

1975

Transactions of the American Mathematical Society

View full text Add to dashboard Cite

ABSTRACT. The space H(E) of holomorphic functions on a quasicomplete nuclear space is investigated. If H(E) is endowed with the compact open topology, it is shown that H(E) is nuclear if and only iff' (continuous dual of E) is nuclear. If E is a VFN (dual of a Fréchet nuclear space) and F is a closed subspace of E, then the restriction mapping H(E) -* H(F) is a surjective strict morphism. Topologies on H(E). In this article, E will denote a quasi-complete nuclear locally convex space over the complex numbers C E' will denote the continuous dual of E endowed with the strong topology (topology of uniform convergence on bounded subsets of E). In a quasi-complete nuclear space E, bounded subsets are relatively compact. Therefore the strong topology on E' coincides with the Mackey topology (topology of uniform convergence on convex compact subsets of E). If U is a convex balanced neighborhood of 0 in E, pv will denote its Minkowski functional.If AT is a convex balanced compact subset of E, [K] will denote the linear span of K in E. IfxG [K], then pK(x) = inf{ X.xGXK and X > 0}. The polar K° ofK is the set {ip: ¡pGE' where \ip\K = sup^^-l^*)! < 1}. As K varies through the convex balanced compact subsets of E, the sets K° form a base of neighborhoods for 0 in E'. pRo is the seminorm defined on E' by pK°(.E' is a nuclear space if given any convex balanced compact subset K of E, there exists another convex balanced compact subset Kx of E containing K such that the mapping E'(K°x) -*■ E'(K°) is a nuclear mapping between normed spaces. E'(K\) -> E'(K°) is nuclear if there exist (an)n Ç [Kx] and (ipn)n ÇE' such that for each ipGE',y = 2Ar> tf/ri)1 'r. Definition 1.1. L(mE), P(mE) will denote, respectively, the continuous w-linear forms and the continuous m-homogeneous polynomials on E. For each

show abstract

On bounded sets of holomorphic mappings

Cited by 15 publications

References 4 publications

Holomorphy on spaces of distribution

Holomorphy on spaces of distribution

Holomorphically Bornological Spaces and Infinite-Dimensional Versions of Hartogs' Theorem

Holomorphic Functions on Nuclear Spaces

Contact Info

Product

Resources

About