scholarly journals Extension of vector-valued functions and weak–strong principles for differentiable functions of finite order

2021 ◽  
Vol 13 (1) ◽  
Author(s):  
Karsten Kruse
Keyword(s):  
Finite Order ◽  
Hausdorff Space ◽  
Linear Operators ◽  
Continuous Linear ◽  
Differentiable Functions ◽  
Weighted Banach Space ◽  
Vector Valued Functions ◽  
Vector Valued ◽  
Locally Convex ◽  
Continuous Linear Operators

AbstractIn this paper we study the problem of extending functions with values in a locally convex Hausdorff space E over a field $$\mathbb {K}$$ K , which has weak extensions in a weighted Banach space $${\mathcal {F}}\nu (\Omega ,\mathbb {K})$$ F ν ( Ω , K ) of scalar-valued functions on a set $$\Omega$$ Ω , to functions in a vector-valued counterpart $$\mathcal {F}\nu (\Omega ,E)$$ F ν ( Ω , E ) of $${\mathcal {F}}\nu (\Omega ,\mathbb {K})$$ F ν ( Ω , K ) . Our findings rely on a description of vector-valued functions as continuous linear operators and extend results of Frerick, Jordá and Wengenroth. As an application we derive weak-strong principles for continuously partially differentiable functions of finite order and vector-valued versions of Blaschke’s convergence theorem for several spaces.

scholarly journals Remarks on completeness in spaces of linear operators

1986 ◽  
Vol 34 (1) ◽  
pp. 25-35
Author(s):  
W. Ricker
Keyword(s):  
Pointwise Convergence ◽  
Hausdorff Space ◽  
Linear Operators ◽  
Continuous Linear ◽  
Mackey Topology ◽  
Sequential Completeness ◽  
Barrelled Spaces ◽  
Topology Of Pointwise Convergence ◽  
Locally Convex ◽  
Continuous Linear Operators

Whereas a locally convex Hausdorff space X inherits any completeness properties that the space of continuous linear operators, L (X), in X, may have (for the topology of pointwise convergence in X), this is not so in the converse situation and is the problem discussed here. The barrelledness of X in its Mackey topology plays an important role: if L (X) is quasicomplete, then X is barrelled for its Mackey topology. Consequently, for Mackey spaces X is turns out that L (X) is quasicomplete if and only if X is quasicomplete and barrelled: this is false if sequential completeness is substituted for quasicompleteness. Furthermore, there exist non-barrelled spaces X for which X and L (X) are quasicomplete (sequentially complete). Hence, although barrelledness is a sufficient condition for completeness of L (X) in various senses, it is certainly not necessary.

Some special operators and new classes of locally convex spaces

1972 ◽  
Vol 71 (3) ◽  
pp. 475-489 ◽  
Author(s):  
Ajit Kaur Chilana
Keyword(s):  
Hausdorff Space ◽  
Linear Operators ◽  
Locally Convex Spaces ◽  
Continuous Linear ◽  
Locally Convex ◽  
Continuous Linear Operators ◽  
Convex Spaces ◽  
Special Classes

AbstractWe consider some special operators on a locally convex Hausdorff space to itself, which have neat spectral theories and prove some perturbation results. This leads us to define and study a few special classes of locally convex spaces in which various subsets of the algebra of continuous linear operators either coincide or are closely related with each other. These are then compared to the classes of barrelled, infrabarrelled and DF-spaces and examples are given to distinguish them from one another.

Strict Topology on Spaces of Continuous Vector-Valued Functions

1979 ◽  
Vol 31 (4) ◽  
pp. 890-896 ◽  
Author(s):  
Seki A. Choo
Keyword(s):  
Tensor Product ◽  
Convex Space ◽  
Hausdorff Space ◽  
Continuous Functions ◽  
Locally Convex Space ◽  
Completely Regular ◽  
Vector Valued Functions ◽  
Bounded Continuous Functions ◽  
Vector Valued ◽  
Locally Convex

In this paper, X denotes a completely regular Hausdorff space, Cb(X) all real-valued bounded continuous functions on X, E a Hausforff locally convex space over reals R, Cb(X, E) all bounded continuous functions from X into E, Cb(X) ⴲ E the tensor product of Cb(X) and E. For locally convex spaces E and F, E ⴲ, F denotes the tensor product with the topology of uniform convergence on sets of the form S X T where S and T are equicontinuous subsets of E′, F′ the topological duals of E, F respectively ([11], p. 96). For a locally convex space G , G ′ will denote its topological dual.

scholarly journals Multipliers of Modules of Continuous Vector-Valued Functions

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
Liaqat Ali Khan ◽  
Saud M. Alsulami
Keyword(s):  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Normed Spaces ◽  
Locally Compact ◽  
Continuous Vector ◽  
Vector Valued Functions ◽  
Vector Valued ◽  
Locally Compact Hausdorff Space ◽  
Locally Convex

In 1961, Wang showed that ifAis the commutativeC*-algebraC0(X)withXa locally compact Hausdorff space, thenM(C0(X))≅Cb(X). Later, this type of characterization of multipliers of spaces of continuous scalar-valued functions has also been generalized to algebras and modules of continuous vector-valued functions by several authors. In this paper, we obtain further extension of these results by showing thatHomC0(X,A)(C0(X,E),C0(X,F))≃Cs,b(X,HomA(E,F)),whereEandFarep-normed spaces which are also essential isometric leftA-modules withAbeing a certain commutativeF-algebra, not necessarily locally convex. Our results unify and extend several known results in the literature.

scholarly journals Operators on Spaces of Bounded Vector-Valued Continuous Functions with Strict Topologies

2015 ◽  
Vol 2015 ◽  
pp. 1-12
Author(s):  
Marian Nowak
Keyword(s):  
Hausdorff Space ◽  
Continuous Functions ◽  
Linear Operators ◽  
Representation Theorems ◽  
Completely Regular ◽  
General Integral ◽  
Continuous Operators ◽  
Bounded Continuous Functions ◽  
Vector Valued ◽  
Continuous Linear Operators

LetXbe a completely regular Hausdorff space, and let(E,‖·‖E)and(F,‖·‖F)be Banach spaces. LetCb(X,E)be the space of allE-valued bounded, continuous functions defined onX, equipped with the strict topologiesβz, where  z=σ,∞,p,τ,t. General integral representation theorems of(βz,‖·‖F)-continuous linear operators  T:Cb(X,E)→F  with respect to the corresponding operator-valued measures are established. Strongly bounded and(βz,‖·‖F)-continuous operatorsT:Cb(X,E)→Fare studied. We extend to “the completely regular setting” some classical results concerning operators on the spacesC(X,E)andCo(X,E), whereX  is a compact or a locally compact space.

Projections and embeddings of locally convex operator spaces and their duals

1984 ◽  
Vol 96 (2) ◽  
pp. 321-323 ◽  
Author(s):  
Jan H. Fourie ◽  
William H. Ruckle
Keyword(s):  
Dual Space ◽  
Sufficient Conditions ◽  
Linear Operators ◽  
Locally Convex Spaces ◽  
Continuous Linear ◽  
Operator Spaces ◽  
Complemented Subspace ◽  
Necessary And Sufficient ◽  
Locally Convex ◽  
Continuous Linear Operators

AbstractLet E, F be Hausdorff locally convex spaces. In this note we consider conditions on E and F such that the dual space of the space Kb (E, F) (of quasi-compact operators) is a complemented subspace of the dual space of Lb (E, F) (of continuous linear operators). We obtain necessary and sufficient conditions for Lb(E, F) to be semi-reflexive.

scholarly journals The inhomogeneous Cauchy-Riemann equation for weighted smooth vector-valued functions on strips with holes

2021 ◽  
Author(s):  
Karsten Kruse
Keyword(s):  
Real Axis ◽  
Hausdorff Space ◽  
Holomorphic Functions ◽  
Smooth Vector ◽  
Riemann Equation ◽  
Dependence Problem ◽  
Cauchy Riemann ◽  
Vector Valued Functions ◽  
Vector Valued ◽  
Locally Convex

AbstractThis paper is dedicated to the question of surjectivity of the Cauchy-Riemann operator $$\overline{\partial }$$ ∂ ¯ on spaces $${\mathcal {E}}{\mathcal {V}}(\varOmega ,E)$$ E V ( Ω , E ) of $${\mathcal {C}}^{\infty }$$ C ∞ -smooth vector-valued functions whose growth on strips along the real axis with holes K is induced by a family of continuous weights $${\mathcal {V}}$$ V . Vector-valued means that these functions have values in a locally convex Hausdorff space E over $${\mathbb {C}}$$ C . We derive a counterpart of the Grothendieck-Köthe-Silva duality $${\mathcal {O}}({\mathbb {C}}\setminus K)/{\mathcal {O}}({\mathbb {C}})\cong {\mathscr {A}}(K)$$ O ( C \ K ) / O ( C ) ≅ A ( K ) with non-empty compact $$K\subset {\mathbb {R}}$$ K ⊂ R for weighted holomorphic functions. We use this duality and splitting theory to prove the surjectivity of $$\overline{\partial }:{\mathcal {E}} {\mathcal {V}}(\varOmega ,E)\rightarrow {\mathcal {E}}{\mathcal {V}} (\varOmega ,E)$$ ∂ ¯ : E V ( Ω , E ) → E V ( Ω , E ) for certain E. This solves the smooth (holomorphic, distributional) parameter dependence problem for the Cauchy-Riemann operator on $${\mathcal {E}}{\mathcal {V}}(\varOmega ,{\mathbb {C}})$$ E V ( Ω , C ) .

scholarly journals Quotient bounded elements in locally convex algebras

1982 ◽  
Vol 33 (1) ◽  
pp. 102-113 ◽  
Author(s):  
Subhash J. Bhatt
Keyword(s):  
Convex Space ◽  
Locally Convex Space ◽  
Linear Operators ◽  
Continuous Linear ◽  
Spatial Characterization ◽  
Measurable Functions ◽  
Locally Convex ◽  
Locally Convex Algebra ◽  
Continuous Linear Operators

AbstractThe quotient bounded and the universally bounded elements in a calibrated locally convex algebra are defined and studied. In the case of a generalized B*-algebra A, they are shown to form respectively b* and B*-algebras, both dense in A. An internal spatial characterization of generalized B*-algebras is obtained. The concepts are illustrated with the help of examples of algebras of measurable functions and of continuous linear operators on a locally convex space.

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