The Weak Basis Theorem Fails in Non-Locally Convex F-Spaces

1977 ◽  
Vol 29 (5) ◽  
pp. 1069-1071 ◽  
Author(s):  
L. Drewnowski
Keyword(s):  
Weak Basis ◽  
Basis Theorem ◽  
Locally Convex

W. J. Stiles showed in [10, Corollary 4.5] that Banach's weak basis theorem fails in the spaces lp, 0 < p < 1. Then, J. H. Shapiro [9] indicated certain general classes of non-locally convex F-spaces with the same property, and asked whether the weak basis theorem fails in every non-locally convex F-space with a weak basis. Our purpose is to answer this question in the affirmative. In [3] we observed that, essentially, the only case that remained open is that of an F-space with irregular basis (en), i.e. such that snen →0 for any scalar sequence (sn).

On the Weak Basis Theorem in F-spaces

1974 ◽  
Vol 26 (6) ◽  
pp. 1294-1300 ◽  
Author(s):  
Joel H. Shapiro
Keyword(s):  
Banach Spaces ◽  
Fréchet Space ◽  
Locally Convex Spaces ◽  
Frechet Space ◽  
Fréchet Spaces ◽  
Weak Basis ◽  
Basis Theorem ◽  
Locally Convex ◽  
Convex Spaces

It is well-known that every weak basis in a Fréchet space is actually a basis. This result, called the weak basis theorem was first given for Banach spaces in 1932 by Banach [1, p. 238], and extended to Fréchet spaces by Bessaga and Petczynski [3]. McArthur [12] proved an analogue for bases of subspaces in Fréchet spaces, and recently W. J. Stiles [18, Corollary 4.5, p. 413] showed that the theorem fails in the non-locally convex spaces lp (0 < p < 1). The purpose of this paper is to prove the following generalization of Stiles' result.

scholarly journals Unconditional convergence and bases

1973 ◽  
Vol 18 (4) ◽  
pp. 321-324 ◽  
Author(s):  
I. Tweddle
Keyword(s):  
Convex Space ◽  
Present Note ◽  
General Setting ◽  
Locally Convex Space ◽  
Unconditional Convergence ◽  
Weak Basis ◽  
Basis Theorem ◽  
Locally Convex ◽  
Extended Basis

The recent papers (6), (7) of J. T. Marti have revived interest in the concept of extended bases, introduced in (1) by M. G. Arsove and R. E. Edwards. In the present note, two results are established which involve this idea. The first of these, which is given in a more general setting, restricts the behaviour of the coefficients for an extended basis in a certain type of locally convex space. The second result extends the well-known weak basis theorem (1, Theorem 11).

scholarly journals The weak basis theorem

1967 ◽  
Vol 17 (1) ◽  
pp. 71-76 ◽  
Author(s):  
Charles McArthur
Keyword(s):  
Weak Basis ◽  
Basis Theorem

Remarks and examples concerning the weak basis theorem

1991 ◽  
Vol 56 (4) ◽  
pp. 376-379
Author(s):  
Jerzy Kakol
Keyword(s):  
Weak Basis ◽  
Basis Theorem

Finite dimensional locally convex cones

Filomat ◽  
2017 ◽  
Vol 31 (16) ◽  
pp. 5111-5116
Author(s):  
Davood Ayaseha
Keyword(s):  
Finite Dimension ◽  
Convex Cones ◽  
Vector Spaces ◽  
Topological Vector Spaces ◽  
Euclidean Topology ◽  
Finite Dimensional ◽  
Dimensional Cone ◽  
Locally Convex Cones ◽  
Locally Convex ◽  
Special Case

We study the locally convex cones which have finite dimension. We introduce the Euclidean convex quasiuniform structure on a finite dimensional cone. In special case of finite dimensional locally convex topological vector spaces, the symmetric topology induced by the Euclidean convex quasiuniform structure reduces to the known concept of Euclidean topology. We prove that the dual of a finite dimensional cone endowed with the Euclidean convex quasiuniform structure is identical with it?s algebraic dual.

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