Linear functionals on vector valued köthe spaces

1974 ◽  
pp. 380-381 ◽  
Author(s):  
C. W. Mullins
Keyword(s):  
Linear Functionals ◽  
Köthe Spaces ◽  
Vector Valued

Representation of Linear Functionals on Köthe Spaces

1953 ◽  
Vol 5 ◽  
pp. 568-575 ◽  
Author(s):  
G. G. Lorentz ◽  
D. G. Wertheim
Keyword(s):  
Banach Space ◽  
Linear Functional ◽  
Continuous Linear ◽  
Linear Functionals ◽  
Continuous Linear Functional ◽  
Köthe Spaces ◽  
Integrable Functions ◽  
Köthe Space ◽  
Vector Valued

Kothe spaces, in the terminology of Diendonné [2], are certain spaces X of real valued integrable functions. In this paper we consider the problem of representation of continuous linear functional on vector valued Kothe spaces. The elements of a Kôthe space X(B) are functions with values in a Banach space B (see §2).

Ordered Vector-Valued Köthe Spaces

1976 ◽  
Vol s2-13 (1) ◽  
pp. 34-40 ◽  
Author(s):  
C. W. Mullins
Keyword(s):  
Köthe Spaces ◽  
Vector Valued

Rotundity In Köthe Spaces of Vector-Valued Functions

1989 ◽  
Vol 41 (4) ◽  
pp. 659-675 ◽  
Author(s):  
A. Kamińska ◽  
B. Turett
Keyword(s):  
Banach Space ◽  
Analogous Result ◽  
Function Spaces ◽  
Sequence Spaces ◽  
Uniform Rotundity ◽  
Köthe Spaces ◽  
Bochner Space ◽  
Köthe Space ◽  
Vector Valued Functions ◽  
Vector Valued

In this paper, Köthe spaces of vector-valued functions are considered. These spaces, which are generalizations of both the Lebesgue-Bochner and Orlicz-Bochner spaces, have been studied by several people (e.g., see [1], [8]). Perhaps the earliest paper concerning the rotundity of such Köthe space is due to I. Halperin [8]. In his paper, Halperin proved that the function spaces E(X) is uniformly rotund exactly when both the Köthe space E and the Banach space X are uniformly rotund; this generalized the analogous result, due to M. M. Day [4], concerning Lebesgue-Bochner spaces. In [20], M. Smith and B. Turett showed that many properties akin to uniform rotundity lift from X to the Lebesgue-Bochner space LP(X) when 1 < p < ∞. A survey of rotundity notions in Lebesgue-Bochner function and sequence spaces can be found in [19].

scholarly journals A Characterization of a Local Vector Valued Bollobás Theorem

2021 ◽  
Vol 76 (4) ◽  
Author(s):  
Sheldon Dantas ◽  
Abraham Rueda Zoca
Keyword(s):  
Type Theorem ◽  
Complete Characterization ◽  
Strict Convexity ◽  
Bilinear Forms ◽  
Linear Functionals ◽  
Projective Tensor Product ◽  
Projective Tensor ◽  
Local Vector ◽  
Vector Valued

AbstractIn this paper, we are interested in giving two characterizations for the so-called property L$$_{o,o}$$ o , o , a local vector valued Bollobás type theorem. We say that (X, Y) has this property whenever given $$\varepsilon > 0$$ ε > 0 and an operador $$T: X \rightarrow Y$$ T : X → Y , there is $$\eta = \eta (\varepsilon , T)$$ η = η ( ε , T ) such that if x satisfies $$\Vert T(x)\Vert > 1 - \eta $$ ‖ T ( x ) ‖ > 1 - η , then there exists $$x_0 \in S_X$$ x 0 ∈ S X such that $$x_0 \approx x$$ x 0 ≈ x and T itself attains its norm at $$x_0$$ x 0 . This can be seen as a strong (although local) Bollobás theorem for operators. We prove that the pair (X, Y) has the L$$_{o,o}$$ o , o for compact operators if and only if so does $$(X, \mathbb {K})$$ ( X , K ) for linear functionals. This generalizes at once some results due to D. Sain and J. Talponen. Moreover, we present a complete characterization for when $$(X \widehat{\otimes }_\pi Y, \mathbb {K})$$ ( X ⊗ ^ π Y , K ) satisfies the L$$_{o,o}$$ o , o for linear functionals under strict convexity or Kadec–Klee property assumptions in one of the spaces. As a consequence, we generalize some results in the literature related to the strongly subdifferentiability of the projective tensor product and show that $$(L_p(\mu ) \times L_q(\nu ); \mathbb {K})$$ ( L p ( μ ) × L q ( ν ) ; K ) cannot satisfy the L$$_{o,o}$$ o , o for bilinear forms.

scholarly journals On a class of vector-valued entire Dirichlet series in n variables

2018 ◽  
Vol 22 (1) ◽  
pp. 3-12 ◽  
Author(s):  
Lakshika Chutani ◽  
Niraj Kumar ◽  
Garima Manocha
Keyword(s):  
Banach Algebra ◽  
Dirichlet Series ◽  
Commutative Banach Algebra ◽  
Continuous Linear ◽  
Linear Functionals ◽  
Entire Dirichlet Series ◽  
Vector Valued ◽  
Class F

We consider a class F of entire Dirichlet series in n variables, whose coefficients belong to a commutative Banach algebra E. With a well defined norm, F is proved to be a Banach algebra with identity. Further results on quasi-invertibility, spectrum and continuous linear functionals are proved for elements belonging to F.

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