The Radon-Nikodym property in conjugate Banach spaces

Stegall, Charles

doi:10.1090/s0002-9947-1975-0374381-1

Cited by 145 publications

(54 citation statements)

References 28 publications

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“…Let F be any separable infinite-dimensional subspace. of E. By a result of Stegall (9), F* is separable. Also, F* is a quotient of E*; hence the proposition applies to E*.…”

Section: Corollary Let E Be An Infinite-dimensional Banach Space Sucmentioning

confidence: 93%

Semi-embeddings of Banach space

Lotz

Peck

Porta

1979

Proceedings of the Edinburgh Mathematical Society

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It is a most implausible fact that a one-to-one operator from c 0 into a Banach space which maps the unit ball of c 0 onto a closed set is necessarily an isomorphism.In this paper the term semi-embedding denotes a one-to-one operator from one Banach space into another, which maps the closed unit ball of the domain onto a closed set. In the first section we study semi-embeddings in conjunction with weak compactness; in the second section we apply our results to the case of semiembeddings defined on C(X), X compact.In Propositions 2 and 3 we construct semi-embeddings which are not isomorphisms. The main result of Section 1 (Proposition 4) states that an operator can be factored through a dual space provided that there exists a semi-embedding defined on the codomain such that the composition of the two operators is weakly compact. This implies that the domain of a weakly compact semi-embedding must be a dual space (Corollary 6). In the case when the domain is C(X), the existence of a weakly compact semi-embedding is equivalent to X being hyperstonian and satisfying the countable chain condition.In Section 2, after some technical lemmas, we strengthen an argument of Pelczynski and Semadeni. Then we give the main result of the paper: a compact space X is scattered if and only if every semi-embedding of C(X) is an isomorphism. This is even true for any equivalent norm on C(X) (Corollary 12). In particular, every semi-embedding of c 0 is an isomorphism.Theorem 11 also allows us to answer a question raised by Kalton and Wilansky (question 6.4 in (5)). Indeed, our results show (see Corollary 14) that a compact space X is scattered if and only if all one-to-one Tauberian operators from C(X) into arbitrary Banach spaces are isomorphisms.Finally, we prove that every semi-embedding of C[0, 1] is an isomorphism on a complemented subspace isomorphic to C[0,1].We use standard terminology throughout (see (1)). An operator is a bounded linear operator from one Banach space into another; an embedding of a Banach space in another is an operator which is an isomorphism onto a closed subspace of its codomain. Finally, a compact space is scattered if it contains no perfect subsets.

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“…Let F be any separable infinite-dimensional subspace. of E. By a result of Stegall (9), F* is separable. Also, F* is a quotient of E*; hence the proposition applies to E*.…”

Section: Corollary Let E Be An Infinite-dimensional Banach Space Sucmentioning

confidence: 93%

Semi-embeddings of Banach space

Lotz

Peck

Porta

1979

Proceedings of the Edinburgh Mathematical Society

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“…a non-empty intersection with a half-space) of arbitrarily small diameter. There are many other equivalent formulations: the reader may refer to [52] or [15] for further enlightenment. Amongst other things, Asplund spaces coincide with spaces whose duals have RNP.…”

Section: Asplund Spaces and The Duality Counterexamplementioning

confidence: 99%

“…This appears implicitly in [52,Corollary 6] (ii) Clearly Y + S 0 /Y is separable, so (i) allows us to choose some separable subspace S of X such that Y + S = Y + S 0 . One may assume that S contains S 0 , replacing S by S + S 0 if necessary.…”

Section: Introductionmentioning

confidence: 99%

Twisted properties of banach spaces

Castillo¹,

Gonzáles²,

Plichko³

et al. 2001

MATH. SCAND.

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If P, Q are two linear topological properties, say that a Banach space X has the property P-by-Q (or is a P-by-Q space) if X has a subspace Y with property P such that the corresponding quotient X/Y has property Q. The choices P, Q ∈ {separable, reflexive} lead naturally to some new results and new proofs of old results concerning weakly compactly generated Banach spaces. For example, every extension of a subspace of L 1 (0, 1) by a WCG space is WCG. They also give a simple new example of a Banach space property which is not a 3-space property but whose dual is a 3-space property.

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“…It is clear that Z is separable, and we may assume that f n ∈ L 1 (µ, Z). Since X * has the RadonNikodým property and Z is a separable subspace of X, we have that Z * is also separable [13]. Since it is obvious that Z * is a norming subspace for Z, we can apply Lemma 3.…”

mentioning

confidence: 95%