Köthe spaces modeled on spaces of $C^∞$ functions

Mefharet Kocatepe; Viacheslav Zahariuta

Displaying similar documents to “Köthe spaces modeled on spaces of $C^{\infty}$ functions”

Geometry of nuclear spaces. II - Linear topological invariants

B. Mityagin (1978-1979)

Séminaire Analyse fonctionnelle (dit "Maurey-Schwartz")

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Compound invariants and embeddings of Cartesian products

P. Chalov, P. Djakov, V. Zahariuta (1999)

Studia Mathematica

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New compound geometric invariants are constructed in order to characterize complemented embeddings of Cartesian products of power series spaces. Bessaga's conjecture is proved for the same class of spaces.

Bessaga's conjecture in unstable Köthe spaces and products

Zefer Nurlu, Jasser Sarsour (1993)

Studia Mathematica

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Let F be a complemented subspace of a nuclear Fréchet space E. If E and F both have (absolute) bases $(e_{n})$ resp. $(f_{n})$ , then Bessaga conjectured (see [2] and for a more general form, also [8]) that there exists an isomorphism of F into E mapping $f_{n}$ to $t_{n} e_{π (k_{n})}$ where $(t_{n})$ is a scalar sequence, π is a permutation of ℕ and $(k_{n})$ is a subsequence of ℕ. We prove that the conjecture holds if E is unstable, i.e. for some base of decreasing zero-neighborhoods $(U_{n})$ consisting of absolutely convex sets one has ∃s ∀p ∃q...

Topological classification of strong duals to nuclear (LF)-spaces

Taras Banakh (2000)

Studia Mathematica

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We show that the strong dual X’ to an infinite-dimensional nuclear (LF)-space is homeomorphic to one of the spaces: $ℝ^{ω}$ , $ℝ^{\infty}$ , $Q \times ℝ^{\infty}$ , $ℝ^{ω} \times ℝ^{\infty}$ , or ${(ℝ^{\infty})}^{ω}$ , where $ℝ^{\infty} = l i m ℝ^{n}$ and $Q = {[- 1, 1]}^{ω}$ . In particular, the Schwartz space D’ of distributions is homeomorphic to ${(ℝ^{\infty})}^{ω}$ . As a by-product of the proof we deduce that each infinite-dimensional locally convex space which is a direct limit of metrizable compacta is homeomorphic either to $ℝ^{\infty}$ or to $Q \times ℝ^{\infty}$ . In particular, the strong dual to any metrizable infinite-dimensional Montel space is homeomorphic...

On Dragilev type power Köthe spaces

P. Djakov, V. Zahariuta (1996)

Studia Mathematica

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A complete isomorphic classification is obtained for Köthe spaces $X = K (e x p [χ (p - κ (i)) - 1 / p] a_{i})$ such that $X q d_{≃} X^{2}$ ; here χ is the characteristic function of the interval [0,∞), the function κ: ℕ → ℕ repeats its values infinitely many times, and $a_{i} \to \infty$ . Any of these spaces has the quasi-equivalence property.

The space of real-analytic functions has no basis

Paweł Domański, Dietmar Vogt (2000)

Studia Mathematica

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Let Ω be an open connected subset of $ℝ^{d}$ . We show that the space A(Ω) of real-analytic functions on Ω has no (Schauder) basis. One of the crucial steps is to show that all metrizable complemented subspaces of A(Ω) are finite-dimensional.

Extendible bases and Kolmogorov problem on asymptotics of entropy and widths of some class of analytic functions

Vyacheslav Zakharyuta (2011)

Annales de la faculté des sciences de Toulouse Mathématiques

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Let $K$ be a compact set in an open set $D$ on a Stein manifold $Ω$ of dimension $n$ . We denote by $H^{\infty} (D)$ the Banach space of all bounded and analytic in $D$ functions endowed with the uniform norm and by $A_{K}^{D}$ a compact subset of the space $C (K)$ consisted of all restrictions of functions from the unit ball $𝔹_{H^{\infty} (D)}$ . In 1950ies Kolmogorov posed a problem: does $ℋ_{ε} (A_{K}^{D}) \sim τ {(ln \frac{1}{ε})}^{n + 1}, ε \to 0,$ where $ℋ_{ε} (A_{K}^{D})$ is the $ε$ -entropy of the compact $A_{K}^{D}$ . We give here a survey of results concerned with this problem and a related problem on...

Tameness in Fréchet spaces of analytic functions

Aydın Aytuna (2016)

Studia Mathematica

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A Fréchet space with a sequence ${| | \cdot | |}_{k}_{k = 1}^{\infty}$ of generating seminorms is called tame if there exists an increasing function σ: ℕ → ℕ such that for every continuous linear operator T from into itself, there exist N₀ and C > 0 such that ${| | T (x) | | ₙ \leq C | | x | |}_{σ (n)}$ ∀x ∈ , n ≥ N₀. This property does not depend upon the choice of the fundamental system of seminorms for and is a property of the Fréchet space . In this paper we investigate tameness in the Fréchet spaces (M) of analytic functions on Stein manifolds M equipped...

Closed universal subspaces of spaces of infinitely differentiable functions

Stéphane Charpentier, Quentin Menet, Augustin Mouze (2014)

Annales de l’institut Fourier

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We exhibit the first examples of Fréchet spaces which contain a closed infinite dimensional subspace of universal series, but no restricted universal series. We consider classical Fréchet spaces of infinitely differentiable functions which do not admit a continuous norm. Furthermore, this leads us to establish some more general results for sequences of operators acting on Fréchet spaces with or without a continuous norm. Additionally, we give a characterization of the existence of a...

Stability of stochastic processes defined by integral functionals

K. Urbanik (1992)

Studia Mathematica

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The paper is devoted to the study of integral functionals $ʃ_{0}^{\infty} f (X (t, ω)) d t$ for continuous nonincreasing functions f and nonnegative stochastic processes X(t,ω) with stationary and independent increments. In particular, a concept of stability defined in terms of the functionals $ʃ_{0}^{\infty} f (a X (t, ω)) d t$ with a ∈ (0,∞) is discussed.

On isomorphic classification of tensor products $E_{\infty} (a) \otimes ̂ E_{\infty}^{'} (b)$

Goncharov A., Zahariuta V., Terzioğlu Tosun

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Abstract New linear topological invariants are introduced and utilized to give an isomorphic classification of tensor products of the type $E_{\infty} (a) \otimes ̂ E_{\infty}^{'} (b)$ , where $E_{\infty} (a)$ is a power series space of infinite type. These invariants are modifications of those suggested earlier by Zahariuta. In particular, some new results are obtained for spaces of infinitely differentiable functions with values in a locally convex space X. These spaces coincide, up to isomorphism, with spaces L(s’,X) of all continuous linear...

Fréchet quotients of spaces of real-analytic functions

P. Domański, L. Frerick, D. Vogt (2003)

Studia Mathematica

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We characterize all Fréchet quotients of the space (Ω) of (complex-valued) real-analytic functions on an arbitrary open set $Ω \subseteq ℝ^{d}$ . We also characterize those Fréchet spaces E such that every short exact sequence of the form 0 → E → X → (Ω) → 0 splits.

Grothendieck-Lidskiĭ theorem for subspaces of quotients of $L_{p}$ -spaces

Oleg Reinov, Qaisar Latif (2014)

Banach Center Publications

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Generalizing A. Grothendieck’s (1955) and V. B. Lidskiĭ’s (1959) trace formulas, we have shown in a recent paper that for p ∈ [1,∞] and s ∈ (0,1] with 1/s = 1 + |1/2-1/p| and for every s-nuclear operator T in every subspace of any $L_{p} (ν)$ -space the trace of T is well defined and equals the sum of all eigenvalues of T. Now, we obtain the analogous results for subspaces of quotients (equivalently: for quotients of subspaces) of $L_{p}$ -spaces.

Asymptotic Vassiliev invariants for vector fields

Sebastian Baader, Julien Marché (2012)

Bulletin de la Société Mathématique de France

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We analyse the asymptotical growth of Vassiliev invariants on non-periodic flow lines of ergodic vector fields on domains of $ℝ^{3}$ . More precisely, we show that the asymptotics of Vassiliev invariants is completely determined by the helicity of the vector field.

On projections of $L^{\infty} (G)$ onto translation-invariant subspaces

W. Chojnacki (1978)

Colloquium Mathematicae

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Stochastic approximation properties in Banach spaces

V. P. Fonf, W. B. Johnson, G. Pisier, D. Preiss (2003)

Studia Mathematica

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We show that a Banach space X has the stochastic approximation property iff it has the stochasic basis property, and these properties are equivalent to the approximation property if X has nontrivial type. If for every Radon probability on X, there is an operator from an $L_{p}$ space into X whose range has probability one, then X is a quotient of an $L_{p}$ space. This extends a theorem of Sato’s which dealt with the case p = 2. In any infinite-dimensional Banach space X there is a compact set K...

Displaying similar documents to “Köthe spaces modeled on spaces of C ∞ functions”

Displaying similar documents to “Köthe spaces modeled on spaces of $C^{\infty}$ functions”