Characterizations of elements of a double dual Banach space and their canonical reproductions

Vassiliki Farmaki

Displaying similar documents to “Characterizations of elements of a double dual Banach space and their canonical reproductions”

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 $(e_{n}^{ζ})$ ,...

Uniqueness of unconditional bases of $c_{0} (l_{p})$ , 0 < p < 1

C. Leránoz (1992)

Studia Mathematica

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We prove that if 0 < p < 1 then a normalized unconditional basis of a complemented subspace of $c_{0} (l_{p})$ must be equivalent to a permutation of a subset of the canonical unit vector basis of $c_{0} (l_{p})$ . In particular, $c_{0} (l_{p})$ has unique unconditional basis up to permutation. Bourgain, Casazza, Lindenstrauss, and Tzafriri have previously proved the same result for $c_{0} (l ₁)$ .

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 $ℓ^{p, \infty}, L^{p, \infty} [0, 1]$ , and $L^{p, \infty} [0, \infty)$ are isomorphic as Banach 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 $D (Ω)$ . ...

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 $ℓ^{1}$ embeds as a complemented subspace of X. Pełczyński proved that every infinite-dimensional closed linear subspace of $ℓ^{1}$ contains a copy of $ℓ^{1}$ that is complemented in $ℓ^{1}$ . Later, Kadec and Pełczyński proved that every non-reflexive closed linear subspace of $L^{1} [0, 1]$ contains a copy of $ℓ^{1}$ 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...

L-summands in their biduals have Pełczyński's property (V*)

Hermann Pfitzner (1993)

Studia Mathematica

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Banach spaces which are L-summands in their biduals - for example $l^{1}$ , the predual of any von Neumann algebra, or the dual of the disc algebra - have Pełczyński’s property (V*), which means that, roughly speaking, the space in question is either reflexive or is weakly sequentially complete and contains many complemented copies of $l^{1}$ .

On spreading $c_{0}$ -sequences in Banach spaces

Vassiliki Farmaki (1999)

Studia Mathematica

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We introduce and study the spreading-(s) and the spreading-(u) property of a Banach space and their relations. A space has the spreading-(s) property if every normalized weakly null sequence has a subsequence with a spreading model equivalent to the usual basis of $c_{0}$ ; while it has the spreading-(u) property if every weak Cauchy and non-weakly convergent sequence has a convex block subsequence with a spreading model equivalent to the summing basis of $c_{0}$ . The main results proved are the...

Estimates of Fourier transforms in Sobolev spaces

V. Kolyada (1997)

Studia Mathematica

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We investigate the Fourier transforms of functions in the Sobolev spaces $W_{1}^{r_{1}, . . ., r_{n}}$ . It is proved that for any function $f \in W_{1}^{r_{1}, . . ., r_{n}}$ the Fourier transform f̂ belongs to the Lorentz space $L^{n / r, 1}$ , where $r = n {(\sum_{j = 1}^{n} 1 / r_{j})}^{- 1} \leq n$ . Furthermore, we derive from this result that for any mixed derivative $D^{s} f (f \in C_{0}^{\infty}, s = (s_{1}, . . ., s_{n}))$ the weighted norm $∥ {(D^{s} f)}^{\land} ∥_{L^{1} (w)} (w (ξ) = {| ξ |}^{- n})$ can be estimated by the sum of $L^{1}$ -norms of all pure derivatives of the same order. This gives an answer to a question posed by A. Pełczyński and M. Wojciechowski.

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 $L_{0} / E$ .

On Banach spaces X for which $Π_{2} (ℒ_{\infty}, X) = B (ℒ_{\infty}, X)$

Ed Dubinsky, A. Pełczyński, H. Rosenthal (1972)

Studia Mathematica

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Banach spaces with a supershrinking basis

Ginés López (1999)

Studia Mathematica

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We prove that a Banach space X with a supershrinking basis (a special type of shrinking basis) without $c_{0}$ copies is somewhat reflexive (every infinite-dimensional subspace contains an infinite-dimensional reflexive subspace). Furthermore, applying the $c_{0}$ -theorem by Rosenthal, it is proved that X contains order-one quasireflexive subspaces if X is not reflexive. Also, we obtain a characterization of the usual basis in $c_{0}$ .

A normalized weakly null sequence with no shrinking subsequence in a Banach space not containing $ℓ_{1}$

E. Odell (1980)

Compositio Mathematica

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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 $\sum_{k = 1}^{\infty} c_{k} φ_{k} (x)$ , where $c_{k} \in l_{q}$ for all q>2, with the following properties: 1. For any p ∈ [1,2) and $f \in L_{μ}^{p} [0, 1] = f : ʃ_{0}^{1} {| f (x) |}^{p} μ (x) d x < \infty$ there are numbers $ɛ_{k}$ , k=1,2,…, $ɛ_{k}$ = 1 or 0, such that $l i m_{n \to \infty} ʃ_{0}^{1} {| \sum_{k = 1}^{n} ɛ_{k} c_{k} φ_{k} (x) - f (x) |}^{p} μ (x) d x = 0 .$ 2. For every p ∈ [1,2) and $f \in L_{μ}^{p} [0, 1]$ there are a function $g \in L^{1} [0, 1]$ with g(x) = f(x) on E and numbers $δ_{k}$ , k=1,2,…, $δ_{k} = 1$ or 0,...