Embedding Nuclear Spaces in Products of an Arbitrary Banach Space

Saxon, Stephen A.

doi:10.2307/2037914

Cited by 7 publications

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“…An old and well-known result of Saxon [10] and Valdivia [15] states that every nuclear space embeds as a subspace of a suitable product of an arbitrarily given infinite-dimensional Banach space. On the other hand, Bellenot [1] showed that not all Schwartz spaces can be embedded into powers of l p for p > 1.…”

Section: Introductionmentioning

confidence: 99%

A note on embedding into product spaces

Sofi

2006

Czech Math J

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Section: Introductionmentioning

confidence: 99%

A note on embedding into product spaces

Sofi

2006

Czech Math J

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“…n If E is the nuclear space s of rapidly decreasing sequences and F is any infinitedimensional Banach space with Schauder basis, then for every 0-neighborhood U in E there exists an absolutely convex 0-neighborhood Vc U in E such that /~v is norm-isomorphic to F (see [6]). This result of Saxon was improved later on by Valdivia [8] proving its validity when E is an arbitrary nuclear space and F any infinite-dimensional separable Banach space.…”

Section: O Introductionmentioning

confidence: 99%

On the embedding of λ-nuclear spaces into product of Banach spaces

García-Lafuente

1986

Acta Math Hung

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O. IntroductionIf E is a nuclear space, Grothendieck [1] proved that for every 0-neighborhood U in E, there is an absolutely convex 0-neighborhood V in E, Vc U such that •v is norm-isomorphic to a subspace of l ~ (whatever pE[1, oo)). As usual, Ev is the completion of the linear space I~)=E/A n-iV normed with the gauge of V.n If E is the nuclear space s of rapidly decreasing sequences and F is any infinitedimensional Banach space with Schauder basis, then for every 0-neighborhood U in E there exists an absolutely convex 0-neighborhood Vc U in E such that /~v is norm-isomorphic to F (see [6]). This result of Saxon was improved later on by Valdivia [8] proving its validity when E is an arbitrary nuclear space and F any infinite-dimensional separable Banach space.In the present paper we bring up these results into the context of 2-nuclearity. Namely, we will prove that for certain sequence spaces 4, a Mackey space a can be found satisfying the following condition: If F is any infinite-dimensional Banach space with Schauder basis, for every 0-neighborhood U in o-there is an absolutely convex 0-neighborhood V in o-such that ffv is norm-isomorphic to F. As a consequence we prove an embedding theorem of k-nuclear spaces into some product of any given infinite-dimensional Banach space with Schauder basis. This research, supported by the University of Extremadura, was carried out during the author's visit to the University of Kaiserslautern (F.R.G.). Definitions and previous resultsThe linear sequence spaces 2 we will deal to are assumed throughout to be normal, additive, decreasing rearrangement invariant and such that 2ci q for some q>0 (see [4] for definitions). The diametral dimension of a Hausdorff locally convex space E, denoted by A (E), is the collection of all sequences (6n) of nonnegative numbers such that for every 0-neighborhood U in E there is a 0-neighborhood V in E, VcU such that On(V, U)<=M6n for every nEN and for some M_->0. Here 6n(V, U) is the n-th Kolmogoroff diameter of V with respect to U. This notion of diametral dimension, different of that of Terzioglu [7], fits better in our context of 2-nuclearity which is defined as follows: A Hausdorff locally convex space E is said to be 2-nuclear if for each p>0 the following condition holds: For every 0-neighborhood U in E there is a 0-neighborhood V in E, Vc U such that (~,(V, U))E2P (a sequence (~n) is in 2p iff (~) is in 4).Following [4] we denote by 4 + the subset of all sequences (~,) in 2 such that 1"

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On the embedding of Schwartz spaces into product spaces

Randtke¹

1976

Proc. Amer. Math. Soc.

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Abstract.Every Schwartz space is embeddable into some sufficiently high power E' of a given Banach space E if and only if E contains /"°°u niformly.In [13] Saxon showed that every nuclear space can be embedded in some sufficiently high power of every Banach space. In [2] Diestel and Lohman showed that a locally convex space that is embeddable in some sufficiently high power of every Banach space is a Schwartz space. In [11] we showed that every Schwartz space can be embedded in some sufficiently high power of c0. In [1] Bellenot gave examples of Schwartz spaces that are not embeddable in any power of lp (1 < p < oo). In Theorem 2 below we characterize those Banach spaces E such that every Schwartz space is embeddable in some sufficiently high power of E. In particular, Theorem 2 implies that every Schwartz space is embeddable in some sufficiently high power of every E^-space, and Corollary 5 implies that if {£"} is a family of Banach spaces such that each Ev is an £p -space (1 < />" < oo), then there is a Schwartz space that is not embeddable in any power of I1F".In [12] (and independently in [6]) it was shown that c0, equipped with the topology of uniform convergence on null sequences in the norm dual of c0 is a universal Schwartz space; and in [12] we asked whether every lE^-space, equipped with the topology of uniform convergence on null sequences in the norm dual, is a universal Schwartz space. Theorem 2 below gives an affirmative answer to this question.The proof of our main result (Theorem 2) depends in a very essential way on the important results of Figiel [3] concerning the factorization of compact linear operators through Banach spaces. Parts of the proof of Theorem 2 are similar to some of the proofs appearing in [11] and [12].A linear operator between locally convex spaces is compact if it transforms some neighborhood of 0 into a relatively compact set. A locally convex space E is a Schwartz space if every continuous linear operator from E into a Banach space is compact. If T: E -> F is a compact linear operator between Banach spaces, then (by [15, Theorem 1, p. 76]) there is a null sequence {an} in the norm dual E' of E such that ||Fx|| < sup|| for every x in E; consequently, if E and F are Banach spaces and § denotes the topology on E

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Embedding Nuclear Spaces in Products of an Arbitrary Banach Space

Cited by 7 publications

References 4 publications

A note on embedding into product spaces

A note on embedding into product spaces

On the embedding of λ-nuclear spaces into product of Banach spaces

On the embedding of Schwartz spaces into product spaces

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