On Holomorphy Versus Linearity in Classifying Locally Convex Spaces

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

doi:10.1016/s0304-0208(08)72193-5

Cited by 24 publications

(12 citation statements)

References 6 publications

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“…Since E is polynomially infrabarrelled it will follow as in Section 4 of [14] that s, n, E is infrabarrelled and hence is a Mackey space. Applying Proposition 1 we conclude that E is polynomially Mackey and thus it follows by the proof Proposition 76 of [3] that E is polynomially bornological. g…”

Section: Polynomially Significant Propertiesmentioning

confidence: 74%

Polynomially significant properties and equivalence of topologies on fully nuclear spaces

Boyd

2004

Complex Variables, Theory and Application: An International Jou

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Section: Polynomially Significant Propertiesmentioning

confidence: 74%

Polynomially significant properties and equivalence of topologies on fully nuclear spaces

Boyd

2004

Complex Variables, Theory and Application: An International Jou

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“…(£/) is contained and bounded in some $?,(£/) it follows that the locally bounded subsets of dK{U) are T h -bounded. The T (> -bounded subsets of ffl(U) are locally bounded if U is an open subset of a Frechet space, a Qi&'M space, a strict inductive limit of Frechet-Montel spaces which admits a continuous norm or an open compact surjective limit of St&M spaces [1,5,7,8,9,10,14]. The converse is not true in general, ( [1,5,8,14]).…”

Section: W R (U) = {Fe %(U): \\F\\ V «X For All N)mentioning

confidence: 99%

“…An open subset U of a locally convex space £ is said to be holomorphically infrabarrelled if the r,, bounded subsets of S^(U) are locally bounded. The set U is said to be holomorphically bornological if for any locally convex space F and f:U-*F, Gateaux holomorphic and bounded on compact sets the function / is holomorphic ( [1]). If U is holomorphically bornological, then U is holomorphically infrabarrelled and Mujica and Nachbin [15] show that U holomorphically infrabarrelled^ %){U) is infrabarrelled and U holomorphically bornological ^>^0(t/) is bornological.…”

Section: [15] {X(u) T a ) = ^(T/)' And Locally Bounded Subsets Of mentioning

confidence: 99%

Linearization of holomorphic mappings on fully nuclear spaces with a basis

et al. 1994

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UDM ^^» G(U) G(UHM)The same construction yields a canonical method of identifying ^(U D M) with a subspace of ^(U n N) for M and N finite dimensional subspaces of E with M c N and hence U n^C^t / n M)) is a subspace of $(£/)• Mujica and Nachbin called this subspace %(£/) A-/and obtained a number of results connecting linear properties of %(U) and holomorphic properties of U. In the process they posed two problems (see Example 10) which we show to have negative answers. Our method is to restrict ourselves to the study of holomorphic functions on fully nuclear spaces with a basis and to use the known structure of such spaces (see [4,5,9,11,12] and [10, Chapters 5 and 6]). In this way we obtain a positive result in the spirit of Mujica and Nachbin [15] but with a strictly weaker hypothesis on more specialized spaces and this immediately leads to the desired counterexamples. t S. Dineen would like to thank the Universidad de Valencia for financial support to visit Valencia to undertake the research for this paper.

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“…□ This corollary answers a question of Leopoldo Nachbin. In [1], the corollary is proved for sets B whose size is at least the continuum and is used (in Lemma 19) to give an example of a discontinuous 2-homogeneous complex polynomial on a locally convex complex vector space (with basis B).…”

Section: Corollarymentioning

confidence: 99%

On a problem of L. Nachbin

Jech¹

1980

Proc. Amer. Math. Soc.

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Abstract. If B is an uncountable set then there is a function r:ixi-»Rt for which there is no function t: B -» R + such that r(6" b2) < f(6,) ■ t(b2) for all bx, b2 e B. Here N denotes the set of all positive integers. Note that the theorem fails if B is countable. Let R+ be the set of all nonnegative reals. Corollary.If B is an uncountable set then there is a function r: B X B -> R+ for which there is no function t: B -* R+ such that r(bx, b2) < t(b) ■ t(b2) for all ¿>" b2 £ B.Proof. Replace the function r by its square r2. □ This corollary answers a question of Leopoldo Nachbin. In [1], the corollary is proved for sets B whose size is at least the continuum and is used (in Lemma 19) to give an example of a discontinuous 2-homogeneous complex polynomial on a locally convex complex vector space (with basis B).Proof of theorem. It suffices to prove the theorem for a set B of cardinality N,. Thus let us assume that B is the set of all countable ordinal numbers. Let < be the well ordering of ordinal numbers.If a £ B, let Wa be the set of all ordinal numbers that are smaller than a; each Wa is at most countable. For each a £ B, let ra be some one-to-one mapping of Wa into N. If a, b £ B, we define r(a, b) = ra(b) whenever a > b, and arbitrarily otherwise.I shall show that the function r: B X B -» N satisfies the statement of the theorem. Thus let / be an arbitrary function from B into N. We want to find a,b £ B such that r(a, b) > t(a) and r(a, b) > t(b).Since B is uncountable,2 there is an uncountable subset C of B such that t is constant on C, with value «. Let a £ C be, such that a has infinitely many

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On Holomorphy Versus Linearity in Classifying Locally Convex Spaces

Cited by 24 publications

References 6 publications

Polynomially significant properties and equivalence of topologies on fully nuclear spaces

Polynomially significant properties and equivalence of topologies on fully nuclear spaces

Linearization of holomorphic mappings on fully nuclear spaces with a basis

On a problem of L. Nachbin

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