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Applications of the theory of semi-embeddings to Banach space theory. (English) Zbl 0541.46020

Let X and Y be Banach spaces and \(T:X\to Y\) an operator (bounded linear operator). T is called a semi-embedding if T is one-one and TU is closed, where U is the unit ball of X, equals the set of \(x\in X\) with \(\| x\| \leq 1.\) If there exists a semi-embedding mapping X into Y, it is said that T semi-embeds in Y. T is called an embedding or isomorphism if there is a \(\delta>0\) so that \(\| Tx\| \geq \delta \| x\|\) for all \(x\in X\). Usually the notion of a semi-embedding is much weaker than that of an embedding. At first, the authors of this paper consider the relationship between semi-embeddings and the Radon-Nikodym property (the RNP).
The main theorem is: Let X be a separable Banach space and suppose X semi-embeds in a Banach space with the RNP. Then X has the RNP.
By this consideration, they have the following results:
A: Let X be a separable Banach space. Then \(X^*\) (dual of X) semi- embeds in \(\ell^ 2.\)
B: Let X be a separable Banach space. If X semi-embeds in a separable dual space, then X has the RNP.
They also consider injective bounded linear operators \(T:X\to Y\) to be a \(G_{\delta}\)-embedding i.e. TK is a \(G_{\delta}\)-set for all closed bounded K.
In the second chapter, they treat mainly semi-embeddings of \(L^ 1\) in \(c_ 0.\)
The main theorem is Theorem 2.3: There is no \(G_{\delta}\)-embedding of \(L^ 1\) in \(c_ 0.\)
By this fact, it is easy to deduce the theorem of Menchoff that there exists a singular probability measure on the circle with Fourier coefficients tending to zero.
In the last chapter, they obtain the following results: There is a subspace Y of \(L^ 1\) with Y isomorphic to \(\ell^ 1\) so that \(T| Y\) is an embedding.
Reviewer: S.Koshi

MSC:

46B20 Geometry and structure of normed linear spaces
46B22 Radon-Nikodým, Kreĭn-Milman and related properties
46B25 Classical Banach spaces in the general theory
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