On the representation of the ring of endomorphisms of a conditionally perfect sequence space

1965 ◽  
Vol 161 (2) ◽  
pp. 125-128
Author(s):  
Arne Persson
Keyword(s):  
Sequence Space ◽  
Perfect Sequence ◽  
Ring Of Endomorphisms

scholarly journals On the Dvoretzky-Rogers theorem

1984 ◽  
Vol 27 (2) ◽  
pp. 105-113
Author(s):  
Fuensanta Andreu
Keyword(s):  
Sequence Space ◽  
Normed Space ◽  
Sequence Spaces ◽  
Finite Dimensional ◽  
Banach Sequence Spaces ◽  
Perfect Sequence ◽  
Normal Topology ◽  
Banach Sequence

The classical Dvoretzky-Rogers theorem states that if E is a normed space for which l1(E)=l1{E} (or equivalently , then E is finite dimensional (see [12] p. 67). This property still holds for any lp (l<p<∞) in place of l1 (see [7]p. 104 and [2] Corollary 5.5). Recently it has been shown that this result remains true when one replaces l1 by any non nuclear perfect sequence space having the normal topology (see [14]). In this context, De Grande-De Kimpe [4] gives an extension of the Devoretzky-Rogers theorem for perfect Banach sequence spaces.

Convergence in Sequence Spaces

1958 ◽  
Vol 11 (2) ◽  
pp. 83-85 ◽  
Author(s):  
H. F. Green
Keyword(s):  
Sequence Space ◽  
Sequence Spaces ◽  
The Other ◽  
Perfect Sequence

In a perfect sequence space α, on which a norm is defined, we can consider three types of convergence, namely projective convergence, strong projective convergence and distance convergence. In the space σ∞, when distance is defined in the usual way, the last two types of convergence coincide and are distinct from projective convergence ((2), p. 316). In the space σ1 all three types of convergence coincide ((2), p. 316). It will be shown in this paper that, if distance convergence and projective convergence coincide, then all three types of convergence coincide. It will not be assumed that the limit under one convergence is also the limit under the other convergence.

scholarly journals $ \lambda(P)$-nuclearity of locally convex spaces having generalized bases

2000 ◽  
Vol 31 (1) ◽  
pp. 9-20
Author(s):  
G. M. Deheri
Keyword(s):  
Sequence Space ◽  
Nuclear Power ◽  
Locally Convex Spaces ◽  
Complete Space ◽  
Infinite Type ◽  
Perfect Sequence ◽  
Power Set ◽  
Locally Convex ◽  
Convex Spaces

It has been established that a $DF$-space having a fully-$\lambda(P)$-basis is $\lambda(P)$-nuclear wherein $P$ is a stable nuclear power set of infinite type. It is shown that a barrelled $G_1$-space $\lambda(Q)$ is uniformly $\lambda(P)$-nuclear iff $\{e_i,e_i\}$ is a fully-$\lambda(P)$-basis for $\lambda(Q)$. Suppose $\lambda$ is a $\mu$-perfect sequence space for a perfect sequence space $\mu$ such that there exist $u\in \lambda^\mu$ and $v\in \mu^x$ with $u_i\ge \varepsilon >0$ and $v_i\ge \iota >0$ for some $\varepsilon$ and $\iota$ and for all $i$. Then the following results are found to be true. (i) A sequentially complete space having a fully-$(\lambda,\sigma \mu)$-basis is $\lambda(P)$-nuclear, provided $\mu$ is a $DF$-space in which $\{e_i,e_i\}$ is a semi-$\lambda(P)$-basis. (ii) Suppose $\{e_i,e_i\}$ is a fully-$(\lambda, \sigma \mu)$-basis for a barrelled $G_i$-space $\lambda(Q)$. If $\mu$ is barrelled and $\{e_i,e_i\}$ is a semi-$\lambda(P)$-basis for $\mu$ then $\lambda(Q)$ is uniformly $\lambda(P)$-nuclear. (iii) A $DF$-space with a fully-$(\lambda, \sigma\mu)$-basis is $\lambda(P)$-nuclear wherein $(\lambda,\sigma\mu)$ is barrelled in which $\{e_i,e_i\}$ is a semi-$\lambda(P)$-basis.

On the domain of Riesz mean in the space Ls

Filomat ◽  
2017 ◽  
Vol 31 (4) ◽  
pp. 925-940 ◽  
Author(s):  
Medine Yeşilkayagil ◽  
Feyzi Başar
Keyword(s):  
Banach Space ◽  
Sequence Space ◽  
Double Sequence ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Double Sequences ◽  
Riesz Mean ◽  
Double Sequence Space ◽  
Necessary And Sufficient ◽  
Barrelled Space

Let 0 < s < ?. In this study, we introduce the double sequence space Rqt(Ls) as the domain of four dimensional Riesz mean Rqt in the space Ls of absolutely s-summable double sequences. Furthermore, we show that Rqt(Ls) is a Banach space and a barrelled space for 1 ? s < 1 and is not a barrelled space for 0 < s < 1. We determine the ?- and ?(?)-duals of the space Ls for 0 < s ? 1 and ?(bp)-dual of the space Rqt(Ls) for 1 < s < 1, where ? ? {p, bp, r}. Finally, we characterize the classes (Ls:Mu), (Ls:Cbp), (Rqt(Ls) : Mu) and (Rqt(Ls):Cbp) of four dimensional matrices in the cases both 0 < s < 1 and 1 ? s < 1 together with corollaries some of them give the necessary and sufficient conditions on a four dimensional matrix in order to transform a Riesz double sequence space into another Riesz double sequence space.

The Cesàro–Gamma operator and its associated sequence space

2021 ◽  
Vol 6 (3) ◽  
Author(s):  
Merve İlkhan Kara ◽  
Hadi Roopaei
Keyword(s):  
Sequence Space ◽  
Gamma Operator
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