Displaying similar documents to “Construction of standard exact sequences of power series spaces”

Some characterizations of ultrabornological spaces

Manuel Valdivia (1974)

Annales de l'institut Fourier

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Let U be an infinite-dimensional separable Fréchet space with a topology defined by a family of norms. Let F be an infinite-dimensional Banach space. Then F is the inductive limit of a family of spaces equal to E . The choice of suitable classes of Fréchet spaces allows to give characterizations of ultrabornological spaces using the result above.. Let Ω be a non-empty open set in the euclidean n -dimensional space R n . Then F is the inductive limit of a family of spaces equal to D ( Ω ) . ...

Isomorphism of certain weak L p spaces

Denny Leung (1993)

Studia Mathematica

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It is shown that the weak L p spaces p , , L p , [ 0 , 1 ] , and L p , [ 0 , ) are isomorphic as Banach spaces.

Two-sided estimates of the approximation numbers of certain Volterra integral operators

D. Edmunds, W. Evans, D. Harris (1997)

Studia Mathematica

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We consider the Volterra integral operator T : L p ( + ) L p ( + ) defined by ( T f ) ( x ) = v ( x ) ʃ 0 x u ( t ) f ( t ) d t . Under suitable conditions on u and v, upper and lower estimates for the approximation numbers a n ( T ) of T are established when 1 < p < ∞. When p = 2 these yield l i m n n a n ( T ) = π - 1 ʃ 0 | u ( t ) v ( t ) | d t . We also provide upper and lower estimates for the α and weak α norms of (an(T)) when 1 < α < ∞.

On the representation of functions by orthogonal series in weighted L p spaces

M. Grigorian (1999)

Studia Mathematica

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It is proved that if φ n is a complete orthonormal system of bounded functions and ɛ>0, then there exists a measurable set E ⊂ [0,1] with measure |E|>1-ɛ, a measurable function μ(x), 0 < μ(x) ≤ 1, μ(x) ≡ 1 on E, and a series of the form k = 1 c k φ k ( x ) , where c k l q for all q>2, with the following properties: 1. For any p ∈ [1,2) and f L μ p [ 0 , 1 ] = f : ʃ 0 1 | f ( x ) | p μ ( x ) d x < there are numbers ɛ k , k=1,2,…, ɛ k = 1 or 0, such that l i m n ʃ 0 1 | k = 1 n ɛ k c k φ k ( x ) - f ( x ) | p μ ( x ) d x = 0 . 2. For every p ∈ [1,2) and f L μ p [ 0 , 1 ] there are a function g L 1 [ 0 , 1 ] with g(x) = f(x) on E and numbers δ k , k=1,2,…, δ k = 1 or 0,...

Tauberian theorems for Cesàro summable double sequences

Ferenc Móricz (1994)

Studia Mathematica

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( s j k : j , k = 0 , 1 , . . . ) be a double sequence of real numbers which is summable (C,1,1) to a finite limit. We give necessary and sufficient conditions under which ( s j k ) converges in Pringsheim’s sense. These conditions are satisfied if ( s j k ) is slowly decreasing in certain senses defined in this paper. Among other things we deduce the following Tauberian theorem of Landau and Hardy type: If ( s j k ) is summable (C,1,1) to a finite limit and there exist constants n 1 > 0 and H such that j k ( s j k - s j - 1 , k - s j - 1 , k + s j - 1 , k - 1 ) - H , j ( s j k - s j - 1 , k ) - H and k ( s j k - s j , k - 1 ) - H whenever j , k > n 1 , then...

Two-sided estimates for the approximation numbers of Hardy-type operators in L and L¹

W. Evans, D. Harris, J. Lang (1998)

Studia Mathematica

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In [2] and [3] upper and lower estimates and asymptotic results were obtained for the approximation numbers of the operator T : L p ( + ) L p ( + ) defined by ( T f ) ( x ) v ( x ) ʃ 0 u ( t ) f ( t ) d t when 1 < p < ∞. Analogous results are given in this paper for the cases p = 1,∞ not included in [2] and [3].

On the complemented subspaces of the Schreier spaces

I. Gasparis, D. Leung (2000)

Studia Mathematica

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It is shown that for every 1 ≤ ξ < ω, two subspaces of the Schreier space X ξ generated by subsequences ( e l n ξ ) and ( e m n ξ ) , respectively, of the natural Schauder basis ( e n ξ ) of X ξ are isomorphic if and only if ( e l n ξ ) and ( e m n ξ ) are equivalent. Further, X ξ admits a continuum of mutually incomparable complemented subspaces spanned by subsequences of ( e n ξ ) . It is also shown that there exists a complemented subspace spanned by a block basis of ( e n ξ ) , which is not isomorphic to a subspace generated by a subsequence of ( e n ζ ) ,...

Isomorphy classes of spaces of holomorphic functions on open polydiscs in dual power series spaces

Manfred Scheve (1991)

Studia Mathematica

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Let Λ_R(α) be a nuclear power series space of finite or infinite type with lim_{j→∞} (1/j) log α_j = 0. We consider open polydiscs D_a in Λ_R(α)'_b with finite radii and the spaces H(D_a) of all holomorphic functions on D_a under the compact-open topology. We characterize all isomorphy classes of the spaces {H(D_a) | a ∈ Λ_R(α), a > 0}. In the case of a nuclear power series space Λ₁(α) of finite type we give this characterization in terms of the invariants (Ω̅ ) and (Ω̃ ) known from...

A rigid space admitting compact operators

Paul Sisson (1995)

Studia Mathematica

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A rigid space is a topological vector space whose endomorphisms are all simply scalar multiples of the identity map. The first complete rigid space was published in 1981 in [2]. Clearly a rigid space is a trivial-dual space, and admits no compact endomorphisms. In this paper a modification of the original construction results in a rigid space which is, however, the domain space of a compact operator, answering a question that was first raised soon after the existence of complete rigid...

Mapping Properties of c 0

Paul Lewis (1999)

Colloquium Mathematicae

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Bessaga and Pełczyński showed that if c 0 embeds in the dual X * of a Banach space X, then 1 embeds as a complemented subspace of X. Pełczyński proved that every infinite-dimensional closed linear subspace of 1 contains a copy of 1 that is complemented in 1 . Later, Kadec and Pełczyński proved that every non-reflexive closed linear subspace of L 1 [ 0 , 1 ] contains a copy of 1 that is complemented in L 1 [ 0 , 1 ] . In this note a traditional sliding hump argument is used to establish a simple mapping property of...

A lifting theorem for locally convex subspaces of L 0

R. Faber (1995)

Studia Mathematica

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We prove that for every closed locally convex subspace E of L 0 and for any continuous linear operator T from L 0 to L 0 / E there is a continuous linear operator S from L 0 to L 0 such that T = QS where Q is the quotient map from L 0 to L 0 / E .

( H p , L p ) -type inequalities for the two-dimensional dyadic derivative

Ferenc Weisz (1996)

Studia Mathematica

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It is shown that the restricted maximal operator of the two-dimensional dyadic derivative of the dyadic integral is bounded from the two-dimensional dyadic Hardy-Lorentz space H p , q to L p , q (2/3 < p < ∞, 0 < q ≤ ∞) and is of weak type ( L 1 , L 1 ) . As a consequence we show that the dyadic integral of a ∞ function f L 1 is dyadically differentiable and its derivative is f a.e.

Partial differential operators depending analytically on a parameter

Frank Mantlik (1991)

Annales de l'institut Fourier

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Let P ( λ , D ) = | α | m a α ( λ ) D α be a differential operator with constant coefficients a α depending analytically on a parameter λ . Assume that the family { P( λ ,D) } is of constant strength. We investigate the equation P ( λ , D ) 𝔣 λ g λ where 𝔤 λ is a given analytic function of λ with values in some space of distributions and the solution 𝔣 λ is required to depend analytically on λ , too. As a special case we obtain a regular fundamental solution of P( λ ,D) which depends analytically on λ . This result answers a question of L. Hörmander. ...