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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Applying coordinate products to the topological identification of normed spaces

Authors: Robert Cauty and Tadeusz Dobrowolski
Journal: Trans. Amer. Math. Soc. 337 (1993), 625-649
MSC: Primary 57N17; Secondary 46B99
MathSciNet review: 1210952
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Abstract: Using the ${l^2}$-products we find pre-Hilbert spaces that are absorbing sets for all Borelian classes of order $\alpha \geq 1$. We also show that the following spaces are homeomorphic to $\Sigma ^\infty$, the countable product of the space $\Sigma = \{(x_n) \in R^\infty : (x_n)$ is bounded}: (1) every coordinate product $\prod _C H_n$ of normed spaces $H_n$ in the sense of a Banach space $C$, where each $H_n$ is an absolute $F_{\sigma \delta }$-set and infinitely many of the $H_n$’s are ${Z_\sigma }$-spaces, (2) every function space $\tilde {L}^p = \cap _{p\prime <p}L^{p\prime }$ with the ${L^q}$-topology, $0<q<p \leq \infty$, (3) every sequence space ${\tilde l^p} = { \cap _{p < p\prime }}{l^{p\prime }}$ with the $l^q$-topology, $0 \leq p < q < \infty$. We also note that each additive and multiplicative Borelian class of order $\alpha \geq 2$, each projective class, and the class of nonprojective spaces contain uncountably many topologically different pre-Hilbert spaces which are $Z_\sigma$-spaces.

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Keywords: Coordinate product, pre-Hilbert space, strong <!– MATH $(\mathcal {K},\mathcal {L})$ –> <IMG WIDTH="59" HEIGHT="41" ALIGN="MIDDLE" BORDER="0" SRC="images/img2.gif" ALT="$(\mathcal {K},\mathcal {L})$">-universality, absorbing set, absolute <!– MATH $F_{\sigma \delta }$ –> <IMG WIDTH="37" HEIGHT="38" ALIGN="MIDDLE" BORDER="0" SRC="images/img1.gif" ALT="$F_{\sigma \delta }$">-set
Article copyright: © Copyright 1993 American Mathematical Society