λ-Bases and their applications

Kamthan, P. K.; Gupta, Manika; Sofi, M. A.

doi:10.1016/0022-247x(82)90177-9

Cited by 3 publications

(4 citation statements)

References 13 publications

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“…| Remark (i). The above result generalizes proposition 5-2 of [6], where the same conclusion has been derived under the assumption of A(P)-nuclearity.…”

Section: Ea(p;t)supporting

confidence: 85%

“…To prove the converse of the above theorem, we require the following result whose proof follows mutatis mutandis on lines similar to that of proposition 3-8 of [6]. Before we prove the converse of the Dynin-Mityagin theorem, we prove the much stronger THEOREM 2-6.…”

Section: Ea(p;t)mentioning

confidence: 99%

“…In fact, even the presence of a fully A(P)-base does not suffice to ensure the A(P)-nuclearity, as is testified by considering any nuclear Frechet Co,-space (cf. [6]).…”

Section: Ea(p;t)mentioning

confidence: 99%

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Some criteria for nuclearity

Sofi

1986

Math. Proc. Camb. Phil. Soc.

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For a given locally convex space, it is always of interest to find conditions for its nuclearity. Well known results of this kind – by now already familiar – involve the use of tensor products, diametral dimension, bilinear forms, generalized sequence spaces and a host of other devices for the characterization of nuclear spaces (see [9]). However, it turns out, these nuclearity criteria are amenable to a particularly simple formulation in the setting of certain sequence spaces; an elegant example is provided by the so-called Grothendieck–Pietsch (GP, for short) criterion for nuclearity of a sequence space (in its normal topology) in terms of the summability of certain numerical sequences.

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“…| Remark (i). The above result generalizes proposition 5-2 of [6], where the same conclusion has been derived under the assumption of A(P)-nuclearity.…”

Section: Ea(p;t)supporting

confidence: 85%

Section: Ea(p;t)mentioning

confidence: 99%

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Some criteria for nuclearity

Sofi

1986

Math. Proc. Camb. Phil. Soc.

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“…Pertaining to generalized bases theory in which the associated sequence space λ carries the usual normal topology, the reader is requested to refer [11], [12] and [14]. The deep rooted relation between λ-base and λ-nuclearity presents a pleasant scenario which can be viewed through [11], [12] and [14].…”

Section: Notations and Preliminary Resultsmentioning

confidence: 99%

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

Deheri¹

2000

Tamkang J. Math.

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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.

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λ-Bases and λ-nuclearity

Kamthan

Sofi

1984

Journal of Mathematical Analysis and Applications

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λ-Bases and their applications

Cited by 3 publications

References 13 publications

Some criteria for nuclearity

Some criteria for nuclearity

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

λ-Bases and λ-nuclearity

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