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A New Class of Perfect Fréchet Spaces

✍ Scribed by Fernando Cobos

Publisher
John Wiley and Sons
Year
1985
Tongue
English
Weight
705 KB
Volume
120
Category
Article
ISSN
0025-584X

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✦ Synopsis

This paper deals with a new class of perfect FRECHET spaces which can be obtained by interpolation of echelon spaces: Zp,q[am,n]. We determine the reflexive, XONTXL, SCHWARTZ, totally reflexive, totally YONTEL and nuclear spaces Zp.q[am,n]. We also derive results on closed subspaces of the spaces (Zp,q)(v.

Many authors have worked on the echelon spaces of order p h 1, introduced by G. KOTHE ( p = l ) , J. DIEUDONNJ~: and A. P. GOMES (psi) (see [7], Q 30 and [14], Chap. 11). Generalizations of these spaces have also been extensively studied.

For example, E.DUBINSKY generalized in [2] the construction of the echelon spaces and showed that one obtains all perfect FRECHET spaces in this way. He also studied those perfect PRECHET spaces that are MONTEL spaces. G. CROFTS completed in [l] the work of E. DUBINSHY and characterized the perfect FR~CHET spaces that are SCHWARTZ spaces. Another generalization of the echelon spaces was studied by C. FENS&, E. SCHOCK [3] and J. PRADA-BLANCO [ll] : the (not necessarily locally convex) spaces A:. This paper deals with the spaces Zp,q[am,n] (l-=p-c.o, 1 sqs-), constructed by replacing the space Zp with the LORENTZ sequence space lp,q (see [S], 4.e, [lo],

13.9 and [13], 1.18.3) as defined by echelon space of orderp. The spaces Zp,q[am.n] can be obtained by interpolation of echelon spaces and form it new class of perfect FRECHET spaces. Purthermore the echelon space of order p is equal to Zp,p [um,n].

We determine the reflexive, MONTEL, SCHWARTZ, totally reflexive, totally MONTEL and nuclear spaces Zp,q[um,n]. We also derive results on closed subspaces of the spaces (Zg,q)(N) in the line of those obtained by A. GROTHENDIECX and 31.

V A L D I V ~~ for the spaces (Zp)(w (see [14], Chap. 11, S 5 , 7).

Part of the content of sections 1,3 and 6 has been taken from the author's doctoral thesis, written, under the direction of Professor MANUEL A. FUGAROLAS a t the University of Santiago de Compostela, but the proofs given here of those results are almost always distinct. The problem treated in section 4 has been suggested to us by Professor MANUEL VALDIVIA.

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