Interpolation in Separable Frechet Spaces with Applications to Spaces of Analytic Functions

1975 ◽  
Vol 27 (5) ◽  
pp. 1110-1113 ◽  
Author(s):  
Paul M. Gauthier ◽  
Lee A. Rubel
Keyword(s):  
Sufficient Conditions ◽  
Fréchet Space ◽  
Complex Numbers ◽  
Frechet Space ◽  
Fréchet Spaces ◽  
Continuous Linear Functional ◽  
Interpolating Sequence ◽  
Necessary And Sufficient ◽  
Locally Convex ◽  
Spaces Of Analytic Functions

Let E be a separable Fréchet space, and let E* be its topological dual space. We recall that a Fréchet space is, by definition, a complete metrizable locally convex topological vector space. A sequence {Ln} of continuous linear functional is said to be interpolating if for every sequence {An} of complex numbers, there exists an ƒ ∈ E such that Ln(ƒ) = An for n = 1, 2, 3, … . In this paper, we give necessary and sufficient conditions that {Ln} be an interpolating sequence. They are different from the conditions in [2] and don't seem to be easily interderivable with them.

scholarly journals FRÉCHET FRAMES, GENERAL DEFINITION AND EXPANSIONS

2014 ◽  
Vol 12 (02) ◽  
pp. 195-208 ◽  
Author(s):  
STEVAN PILIPOVIĆ ◽  
DIANA T. STOEVA
Keyword(s):  
Banach Spaces ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Sequence Spaces ◽  
General Definition ◽  
Fréchet Space ◽  
Frechet Space ◽  
Bk Space ◽  
Frame Expansions ◽  
Necessary And Sufficient

We define an (X1, Θ, X2)-frame with Banach spaces X2 ⊆ X1, ‖ ⋅ ‖1 ≤ ‖ ⋅ ‖2, and a BK-space [Formula: see text]. Then by the use of decreasing sequences of Banach spaces [Formula: see text] and of sequence spaces [Formula: see text], we define a General Fréchet frame on the Fréchet space [Formula: see text]. We obtain frame expansions of elements of XF and its dual [Formula: see text], as well of some of the generating spaces of XF with convergence in appropriate norms. Moreover, we determine necessary and sufficient conditions for a General pre-Fréchet frame to be a General Fréchet frame, as well as for the complementedness of the range of the analysis operator U : XF → ΘF. Several examples illustrate our investigations.

scholarly journals The regularity of the space of germs of Fréchet valued holomorphic functions and the mixed Hartog's theorem

2006 ◽  
Vol 99 (1) ◽  
pp. 119 ◽  
Author(s):  
Thai Thuan Quang
Keyword(s):  
Holomorphic Function ◽  
Sufficient Conditions ◽  
Holomorphic Functions ◽  
Schwartz Space ◽  
Necessary And Sufficient Conditions ◽  
Fréchet Space ◽  
Compact Set ◽  
Frechet Space ◽  
Set Of Uniqueness ◽  
Necessary And Sufficient

It is shown that $H(K, F)$ is regular for every reflexive Fréchet space $F$ with the property ($\mathrm{LB}_\infty)$ where $K$ is a compact set of uniqueness in a Fréchet-Schwartz space $E$ such that $E \in (\Omega)$. Using this result we give necessary and sufficient conditions for a Fréchet space $F$, under which every separately holomorphic function on $K \times F^*$ is holomorphic, where $K$ is as above.

On the Weak Basis Theorem in F-spaces

1974 ◽  
Vol 26 (6) ◽  
pp. 1294-1300 ◽  
Author(s):  
Joel H. Shapiro
Keyword(s):  
Banach Spaces ◽  
Fréchet Space ◽  
Locally Convex Spaces ◽  
Frechet Space ◽  
Fréchet Spaces ◽  
Weak Basis ◽  
Basis Theorem ◽  
Locally Convex ◽  
Convex Spaces

It is well-known that every weak basis in a Fréchet space is actually a basis. This result, called the weak basis theorem was first given for Banach spaces in 1932 by Banach [1, p. 238], and extended to Fréchet spaces by Bessaga and Petczynski [3]. McArthur [12] proved an analogue for bases of subspaces in Fréchet spaces, and recently W. J. Stiles [18, Corollary 4.5, p. 413] showed that the theorem fails in the non-locally convex spaces lp (0 < p < 1). The purpose of this paper is to prove the following generalization of Stiles' result.

scholarly journals The existence of a factorized unbounded operator between Fréchet spaces

2018 ◽  
Vol 13 (01) ◽  
pp. 2050017
Author(s):  
Ersin Kızgut ◽  
Murat Yurdakul
Keyword(s):  
Unbounded Operator ◽  
Fréchet Space ◽  
Locally Convex Spaces ◽  
Frechet Space ◽  
Fréchet Spaces ◽  
Continuous Linear ◽  
Linear Map ◽  
Infinite Dimensional ◽  
Locally Convex ◽  
Convex Spaces

For locally convex spaces [Formula: see text] and [Formula: see text], the continuous linear map [Formula: see text] is called bounded if there is a zero neighborhood [Formula: see text] of [Formula: see text] such that [Formula: see text] is bounded in [Formula: see text]. Our main result is that the existence of an unbounded operator [Formula: see text] between Fréchet spaces [Formula: see text] and [Formula: see text] which factors through a third Fréchet space [Formula: see text] ends up with the fact that the triple [Formula: see text] has an infinite dimensional closed common nuclear Köthe subspace, provided that [Formula: see text] has the property [Formula: see text].

scholarly journals Quasi-bounded sets

1990 ◽  
Vol 13 (3) ◽  
pp. 607-610
Author(s):  
Jan Kucera
Keyword(s):  
Fréchet Space ◽  
Locally Convex Spaces ◽  
Frechet Space ◽  
Fréchet Spaces ◽  
Bounded Set ◽  
Bounded Sets ◽  
Locally Convex ◽  
Convex Spaces

It is proved in [1] & [2] that a set bounded in an inductivelimit E=indlim Enof Fréchet spaces is also bounded in someEniffEis fast complete. In the case of arbitrary locally convex spacesEnevery bounded set in a fast completeindlim Enis quasi-bounded in someEn, though it may not be bounded or even contained in anyEn. Every bounded set is quasi-bounded. In a Fréchet space every quasi-bounded set is also bounded.

scholarly journals QKtype spaces of analytic functions

2006 ◽  
Vol 4 (1) ◽  
pp. 73-84 ◽  
Author(s):  
Hasi Wulan ◽  
Jizhen Zhou
Keyword(s):  
Unit Disk ◽  
Analytic Functions ◽  
Sufficient Conditions ◽  
Nondecreasing Function ◽  
Necessary And Sufficient Conditions ◽  
Type Space ◽  
Necessary And Sufficient ◽  
Spaces Of Analytic Functions

For a nondecreasing functionK:[0,8)?[0,8)and0<p<8,-2<q<8, we introduceQK(p,q), aQKtype space of functions analytic in the unit disk and study the characterizations ofQK(p,q). Necessary and sufficient conditions onKsuch thatQK(p,q)become some known spaces are given.

A class of simple non-weight modules over the virasoro algebra

2020 ◽  
Vol 63 (4) ◽  
pp. 956-970 ◽  
Author(s):  
Haibo Chen ◽  
JianZhi Han
Keyword(s):  
Tensor Product ◽  
Sufficient Conditions ◽  
Virasoro Algebra ◽  
Vector Spaces ◽  
Complex Numbers ◽  
Infinite Dimensional ◽  
Highest Weight ◽  
Alpha Omega ◽  
Necessary And Sufficient ◽  
Infinite Dimensional Lie Algebra

AbstractThe Virasoro algebra $\mathcal {L}$ is an infinite-dimensional Lie algebra with basis {Lm, C| m ∈ ℤ} and relations [Lm, Ln] = (n − m)Lm+n + δm+n,0((m3 − m)/12)C, [Lm, C] = 0 for m, n ∈ ℤ. Let $\mathfrak a$ be the subalgebra of $\mathcal {L}$ spanned by Li for i ≥ −1. For any triple (μ, λ, α) of complex numbers with μ ≠ 0, λ ≠ 0 and any non-trivial $\mathfrak a$-module V satisfying the condition: for any v ∈ V there exists a non-negative integer m such that Liv = 0 for all i ≥ m, non-weight $\mathcal {L}$-modules on the linear tensor product of V and ℂ[∂], denoted by $\mathcal {M}(V,\mu ,\Omega (\lambda ,\alpha ))\ (\Omega (\lambda ,\alpha )=\mathbb {C}[\partial ]$ as vector spaces), are constructed in this paper. We prove that $\mathcal {M}(V,\mu ,\Omega (\lambda ,\alpha ))$ is simple if and only if μ ≠ 1, λ ≠ 0, α ≠ 0. We also give necessary and sufficient conditions for two such simple $\mathcal {L}$-modules being isomorphic. Finally, these simple $\mathcal {L}$-modules $\mathcal {M}(V,\mu ,\Omega (\lambda ,\alpha ))$ are proved to be new for V not being the highest weight $\mathfrak a$-module whose highest weight is non-zero.

Local analytic structure in certain commutative topological algebras

1972 ◽  
Vol 6 (2) ◽  
pp. 161-167 ◽  
Author(s):  
R.J. Loy
Keyword(s):  
Power Series ◽  
Topological Algebra ◽  
Fréchet Space ◽  
Analytic Structure ◽  
Frechet Space ◽  
Topological Algebras ◽  
Space Topology ◽  
Formal Power ◽  
Locally Convex Topology ◽  
Locally Convex

Let B be a topological algebra with Fréchet space topology, A an algebra with locally convex topology and an algebra of formal power series over A in n commuting indeterminates which carries a Fréchet space topology. In a previous paper the author showed, for the case n = 1, that a homomorphism of B into whose range contains polynomials is necessarily continuous provided the coordinate projections of into A satisfy a certain equicontinuity condition. This result is here extended to the case of general n, and also to weaker topological assumptions.

On the Construction of Nonnegative 5×5 Matrices from Spectrum Data

2013 ◽  
Vol 444-445 ◽  
pp. 621-624
Author(s):  
Zhi Bing Liu ◽  
Zhen Tu ◽  
Cheng Feng Xu
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Complex Numbers ◽  
Nonnegative Matrices ◽  
Necessary And Sufficient ◽  
Spectrum Data

This paper studies the construction problems of five order nonnegative matrices from spectrum data. Let be a list of complex numbers with . Necessary and sufficient conditions for the existence of an entry-wise nonnegative 5×5 matrix with spectrum are presented.

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