The Mackey topology and complemented subspaces of Lorentz sequence spaces for

Authors:
M. Nawrocki and A. Ortyński

Journal:
Trans. Amer. Math. Soc. **287** (1985), 713-722

MSC:
Primary 46A45; Secondary 46A10

DOI:
https://doi.org/10.1090/S0002-9947-1985-0768736-7

MathSciNet review:
768736

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we continue the study of Lorentz sequence spaces , , initiated by N. Popa [**8**]. First we show that the Mackey completion of is equal to for some sequence . Next, we prove that if , then it contains a complemented subspace isomorphic to . Finally we show that if , then every complemented subspace of with symmetric bases is isomorphic to .

**[1]**Z. Altshuler, P. G. Cassaza and B. L. Lin,*On symmetric basic sequences in Lorentz sequence spaces*, Israel J. Math.**15**(1973), 140-155. MR**0328553 (48:6895)****[2]**P. G. Cassaza and B. L. Lin,*On symmetric basic sequences in Lorentz sequence spaces*. II, Israel J. Math.**17**(1974), 191-218. MR**0348443 (50:941)****[3]**-,*Projections in Banach spaces with symmetric bases*, Studia Math.**52**(1974), 189-194. MR**0350384 (50:2877)****[4]**D. J. H. Garling,*On symmetric sequence spaces*, Proc. London Math. Soc.**16**(1966), 85-106. MR**0192311 (33:537)****[5]**H. E. Lacey,*The isometric theory of classical Banach spaces*, Springer-Verlag, Berlin and New York, 1974. MR**0493279 (58:12308)****[6]**J. Lindenstrauss and L. Tzafriri,*Classical Banach spaces*. I.*Sequences spaces*, Springer-Verlag, Berlin and New York, 1977. MR**0500056 (58:17766)****[7]**A. Ortyński,*Projections in locally bounded spaces with symmetric bases*, Comment. Math. (to appear).**[8]**N. Popa,*Basic sequences and subspaces in Lorentz sequence spaces without local convexity*, Trans. Amer. Math. Soc.**263**(1981), 431-456. MR**594418 (82f:46012)****[9]**S. Rolewicz,*Metric linear spaces*, PWN, Warsaw, 1972. MR**808176 (88i:46004b)****[10]**J. H. Shapiro,*Mackey topologies, reproducing kernels, and diagonal maps on Hardy and Bergman spaces*, Duke Math. J.**43**(1976), 187-202. MR**0500100 (58:17806)**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
46A45,
46A10

Retrieve articles in all journals with MSC: 46A45, 46A10

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1985-0768736-7

Keywords:
-Banach spaces,
Mackey topology,
complemented subspaces

Article copyright:
© Copyright 1985
American Mathematical Society