scholarly journals On the Projective Description of Weighted (LF)-Spaces of Continuous Functions

2014 ◽  
Vol 2014 ◽  
pp. 1-5
Author(s):  
Catherine V. Komarchuk ◽  
Sergej N. Melikhov
Keyword(s):  
Fourier Analysis ◽  
Continuous Functions ◽  
Fréchet Spaces ◽  
Inductive Limits ◽  
Homogeneous Functions ◽  
Spaces Of Continuous Functions ◽  
Algebraic Description ◽  
Projective Description

We solve the problem of the topological or algebraic description of countable inductive limits of weighted Fréchet spaces of continuous functions on a cone. This problem is investigated for two families of weights defined by positively homogeneous functions. Weights of this form play the important role in Fourier analysis.

scholarly journals Distinguishedness of weighted Fréchet spaces of continuous functions

1992 ◽  
Vol 35 (2) ◽  
pp. 271-283 ◽  
Author(s):  
Françoise Bastin
Keyword(s):  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Continuous Functions ◽  
Fréchet Space ◽  
Frechet Space ◽  
Fréchet Spaces ◽  
Density Condition ◽  
Locally Compact ◽  
Spaces Of Continuous Functions ◽  
Locally Compact Hausdorff Space

In this paper, we prove that if is an increasing sequence of strictly positive and continuous functions on a locally compact Hausdorff space X such that then the Fréchet space C(X) is distinguished if and only if it satisfies Heinrich's density condition, or equivalently, if and only if the sequence satisfies condition (H) (cf. e.g.‵[1] for the introduction of (H)). As a consequence, the bidual λ∞(A) of the distinguished Köthe echelon space λ0(A) is distinguished if and only if the space λ1(A) is distinguished. This gives counterexamples to a problem of Grothendieck in the context of Köthe echelon spaces.

scholarly journals On weighted inductive limits of spaces of Fréchet-valued continuous functions

1991 ◽  
Vol 50 (2) ◽  
pp. 233-242 ◽  
Author(s):  
Antonio Galbis
Keyword(s):  
Continuous Functions ◽  
Linear Mapping ◽  
Fréchet Spaces ◽  
Continuous Linear ◽  
Inductive Limits ◽  
Continuous Linear Mapping ◽  
Weighted Inductive Limits

AbstractIn this article we continue the study of weighted inductive limits of spaces of Fréchet-valued continuous functions, concentrating on the problem of projective descriptions and the barrelledness of the corresponding “projective hull”. Our study is related to the work of Vogt on the study of pairs (E, F) of Fréchet spaces such that every continuous linear mapping from E into F is bounded and on the study of the functor Ext1 (E, F) for pairs (E, F) of Fréchet spaces.

scholarly journals SUBPROJECTIVITY OF PROJECTIVE TENSOR PRODUCTS OF BANACH SPACES OF CONTINUOUS FUNCTIONS

2021 ◽  
pp. 1-14
Author(s):  
R.M. CAUSEY
Keyword(s):  
Tensor Product ◽  
Banach Spaces ◽  
Tensor Products ◽  
Continuous Functions ◽  
Hausdorff Spaces ◽  
Projective Tensor Product ◽  
Projective Tensor ◽  
Spaces Of Continuous Functions

Abstract Galego and Samuel showed that if K, L are metrizable, compact, Hausdorff spaces, then $C(K)\widehat{\otimes}_\pi C(L)$ is c0-saturated if and only if it is subprojective if and only if K and L are both scattered. We remove the hypothesis of metrizability from their result and extend it from the case of the twofold projective tensor product to the general n-fold projective tensor product to show that for any $n\in\mathbb{N}$ and compact, Hausdorff spaces K1, …, K n , $\widehat{\otimes}_{\pi, i=1}^n C(K_i)$ is c0-saturated if and only if it is subprojective if and only if each K i is scattered.

scholarly journals BOUNDED LINEAR OPERATORS ON SPACES IN NORMED DUALITY

2007 ◽  
Vol 49 (1) ◽  
pp. 145-154
Author(s):  
BRUCE A. BARNES
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Bounded Linear Operator ◽  
Integral Operators ◽  
Continuous Functions ◽  
Linear Operators ◽  
Bounded Linear Operators ◽  
Closed Range ◽  
Bounded Linear ◽  
Spaces Of Continuous Functions

Abstract.LetTbe a bounded linear operator on a Banach spaceW, assumeWandYare in normed duality, and assume thatThas adjointT†relative toY. In this paper, conditions are given that imply that for all λ≠0, λ−Tand λ −T†maintain important standard operator relationships. For example, under the conditions given, λ −Thas closed range if, and only if, λ −T†has closed range.These general results are shown to apply to certain classes of integral operators acting on spaces of continuous functions.

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