A ν-Integral Representation for the Continuous Linear Operators on Spaces of Continuously Differentiable Vector-Valued Functions

1971 ◽  
Vol 30 (2) ◽  
pp. 263
Author(s):  
J. R. Edwards ◽  
S. G. Wayment
Keyword(s):  
Integral Representation ◽  
Linear Operators ◽  
Continuous Linear ◽  
Vector Valued Functions ◽  
Vector Valued ◽  
Differentiable Vector ◽  
Continuous Linear Operators

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 Integration in Orlicz-Bochner Spaces

2018 ◽  
Vol 2018 ◽  
pp. 1-9
Author(s):  
Marian Nowak
Keyword(s):  
Integral Representation ◽  
Measure Space ◽  
Finite Measure ◽  
Linear Operators ◽  
Continuous Linear ◽  
Topological Properties ◽  
Young Function ◽  
Finite Measure Space ◽  
Bochner Space ◽  
Continuous Linear Operators

Let (Ω,Σ,μ) be a complete σ-finite measure space, φ be a Young function, and X and Y be Banach spaces. Let Lφ(X) denote the Orlicz-Bochner space, and Tφ∧ denote the finest Lebesgue topology on Lφ(X). We study the problem of integral representation of (Tφ∧,·Y)-continuous linear operators T:Lφ(X)→Y with respect to the representing operator-valued measures. The relationships between (Tφ∧,·Y)-continuous linear operators T:Lφ(X)→Y and the topological properties of their representing operator measures are established.

An operator version of Abel's continuity theorem

2010 ◽  
Vol 17 (4) ◽  
pp. 787-794
Author(s):  
Vaja Tarieladze
Keyword(s):  
Banach Space ◽  
Linear Operators ◽  
Identity Operator ◽  
Continuous Linear ◽  
Continuity Theorem ◽  
Operator Version ◽  
Continuous Linear Operators

Abstract For a Banach space X let 𝔄 be the set of continuous linear operators A : X → X with ‖A‖ < 1, I be the identity operator and 𝔄 c ≔ {A ∈ 𝔄 : ‖I – A‖ ≤ c(1 – ‖A‖)}, where c ≥ 1 is a constant. Let, moreover, (xk ) k≥0 be a sequence in X such that the series converges and ƒ : 𝔄 ∪ {I} → X be the mapping defined by the equality It is shown that ƒ is continuous on 𝔄 and for every c ≥ 1 the restriction of ƒ to 𝔄 c ∪ {I} is continuous at I.

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