Operator and dual operator bases in linear topological spaces

Author:
William B. Johnson

Journal:
Trans. Amer. Math. Soc. **166** (1972), 387-400

MSC:
Primary 46A15

DOI:
https://doi.org/10.1090/S0002-9947-1972-0296643-6

MathSciNet review:
0296643

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: A net of continuous linear projections of finite range on a Hausdorff linear topological space *V* is said to be a Schauder operator basis--S.O.B. --(resp. Schauder dual operator basis--S.D.O.B.) provided it is pointwise bounded and converges pointwise to the identity operator on *V*, and (resp. ) whenever .

S.O.B.'s and S.D.O.B.'s are natural generalizations of finite dimensional Schauder bases of subspaces. In fact, a sequence of operators is both a S.O.B. and S.D.O.B. iff it is the sequence of partial sum operators associated with a finite dimensional Schauder basis of subspaces.

We show that many duality-theory results concerning Schauder bases can be extended to S.O.B.'s or S.D.O.B.'s. In particular, a space with a S.D.O.B. is semi-reflexive if and only if the S.D.O.B. is shrinking and boundedly complete.

Several results on S.O.B.'s and S.D.O.B.'s were previously unknown even in the case of Schauder bases. For example, Corollary IV.2 implies that the strong dual of an evaluable space which admits a shrinking Schauder basis is a complete barrelled space.

**[1]**M. G. Arsove,*The Payley-Wiener theorem in metric linear spaces*, Pacific J. Math.**10**(1960), 365-379. MR**23**#A2731. MR**0125429 (23:A2731)****[2]**C. Bessaga and A. Pełczyński,*Properties of bases in spaces of type*, Prace Mat.**3**(1959), 123-142. (Polish) MR**23**#A3986. MR**0126691 (23:A3986)****[3]**J. Dieudonné,*On biorthogonal systems*, Michigan Math. J.**2**(1954), 7-20. MR**16**, 47. MR**0062946 (16:47c)****[4]**E. Dubinsky and J. R. Retherford,*Schauder bases and Köthe sequence spaces*, Trans. Amer. Math. Soc.**130**(1968), 265-280. MR**38**#510. MR**0232184 (38:510)****[5]**J. A. Dyer,*Integral bases in linear topological spaces*, Illinois J. Math.**14**(1970), 468-477. MR**41**#5923. MR**0261308 (41:5923)****[6]**M. M. Grinblyum,*On the representation of a space of type B in the form of a direct sum of subspaces*, Dokl. Akad. Nauk SSSR**70**(1950), 749-752. (Russian) MR**11**, 525. MR**0033977 (11:525c)****[7]**R. C. James,*Bases and reflexivity of Banach spaces*, Ann. of Math. (2)**52**(1950), 518-527. MR**12**, 616. MR**0039915 (12:616b)****[8]**O. T. Jones,*Continuity of seminorms and linear mappings on a space with Schauder basis*, Studia Math.**34**(1970), 121-126. MR**41**#4182. MR**0259544 (41:4182)****[9]**J. L. Kelley and I. Namioka,*Linear topological spaces*, The University Series in Higher Math., Van Nostrand, Princeton, N. J., 1963. MR**29**#3851. MR**0166578 (29:3851)****[10]**C. W. McArthur,*The weak basis theorem*, Colloq. Math.**17**(1967), 71-76. MR**35**#7103. MR**0216268 (35:7103)****[11]**J. R. Retherford,*Bases, basic sequences and reflexivity of linear topological spaces*, Math. Ann.**164**(1966), 280-285. MR**33**#6351. MR**0198192 (33:6351)****[12]**W. H. Ruckle,*The infinite sum of closed subspaces of an F-space*, Duke Math. J.**31**(1964), 543-554. MR**29**#3862. MR**0166589 (29:3862)****[13]**I. Singer,*Basic sequences and reflexivity of Banach spaces*, Studia Math.**21**(1961/62), 351-369. MR**26**#4155. MR**0146635 (26:4155)**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
46A15

Retrieve articles in all journals with MSC: 46A15

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1972-0296643-6

Keywords:
Schauder basis,
continuous linear projections,
duality

Article copyright:
© Copyright 1972
American Mathematical Society