scholarly journals The product of vector-valued measures

1973 ◽  
Vol 8 (3) ◽  
pp. 359-366 ◽  
Author(s):  
Charles Swartz
Keyword(s):  
Scalar Field ◽  
Banach Spaces ◽  
Vector Measure ◽  
Sufficient Condition ◽  
Pettis Integral ◽  
Necessary And Sufficient Condition ◽  
Bilinear Map ◽  
Scalar Measure ◽  
Necessary And Sufficient ◽  
Vector Valued

Let M (N) be a σ–algebra of subsets of a set S (T) and let X, Y be Banach spaces with (,) a continuous bilinear map from X × Y into the scalar field. If μ: M → X (v: N → Y) is a vector measure and λ is the scalar measure defined on the measurable rectangles A × B, A ∈ M, B ∈ N, by λ(A×B) = 〈μ(A), v(B)〉, it is known that λ is generally not countably additive on the algebra generated by the measurable rectangles and therefore has no countably additive extension to the σ-algebra generated by the measurable rectangles. If μ (v) is an indefinite Pettis integral it is shown that a necessary and sufficient condition that λ have a countable additive extension to the σ-algebra generated by the measurable rectangles is that the function F: (s, t) → 〈f(s), g(t)〉 is integrable with respect to α × β.

scholarly journals Semi-smooth Points in Some Classical Function Spaces

2021 ◽  
Vol 77 (1) ◽  
Author(s):  
Beata Derȩgowska ◽  
Beata Gryszka ◽  
Karol Gryszka ◽  
Paweł Wójcik
Keyword(s):  
Continuous Function ◽  
Metric Space ◽  
Continuous Functions ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Compact Metric Space ◽  
Spaces Of Continuous Functions ◽  
Classical Function ◽  
Necessary And Sufficient ◽  
Vector Valued

AbstractThe investigations of the smooth points in the spaces of continuous function were started by Banach in 1932 considering function space $$\mathcal {C}(\Omega )$$ C ( Ω ) . Singer and Sundaresan extended the result of Banach to the space of vector valued continuous functions $$\mathcal {C}(\mathcal {T},E)$$ C ( T , E ) , where $$\mathcal {T}$$ T is a compact metric space. The aim of this paper is to present a description of semi-smooth points in spaces of continuous functions $$\mathcal {C}_0(\mathcal {T},E)$$ C 0 ( T , E ) (instead of smooth points). Moreover, we also find necessary and sufficient condition for semi-smoothness in the general case.

ON BI-BANACH FRAMES IN BANACH SPACES

2014 ◽  
Vol 12 (02) ◽  
pp. 1450015 ◽  
Author(s):  
SHALU SHARMA
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Banach Frames ◽  
Banach Frame ◽  
Necessary And Sufficient

Bi-Banach frames in Banach spaces have been defined and studied. A necessary and sufficient condition under which a Banach space has a Bi-Banach frame has been given. Finally, Pseudo exact retro Banach frames have been defined and studied.

scholarly journals Some results concerning frames in Banach spaces

2007 ◽  
Vol 38 (3) ◽  
pp. 267-276 ◽  
Author(s):  
S. K. Kaushik
Keyword(s):  
Finite Number ◽  
Banach Spaces ◽  
Necessary Condition ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Linearly Independent ◽  
Banach Frame ◽  
Minimal Sequence ◽  
Definition Of ◽  
Necessary And Sufficient

A necessary and sufficient condition for the associated sequence of functionals to a complete minimal sequence to be a Banach frame has been given. We give the definition of a weak-exact Banach frame, and observe that an exact Banach frame is weak-exact. An example of a weak-exact Banach frame which is not exact has been given. A necessary and sufficient condition for a Banach frame to be a weak-exact Banach frame has been obtained. Finally, a necessary condition for the perturbation of a retro Banach frame by a finite number of linearly independent vectors to be a retro Banach frame has been given.

Vector-valued nonuniform multiresolution analysis on local fields

2015 ◽  
Vol 13 (04) ◽  
pp. 1550029 ◽  
Author(s):  
Firdous Ahmad Shah ◽  
M. Younus Bhat
Keyword(s):  
Multiresolution Analysis ◽  
Orthonormal Basis ◽  
Positive Characteristic ◽  
Local Fields ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Vilenkin Groups ◽  
Refinement Mask ◽  
Necessary And Sufficient ◽  
Vector Valued

A multiresolution analysis (MRA) on local fields of positive characteristic was defined by Shah and Abdullah for which the translation set is a discrete set which is not a group. In this paper, we continue the study based on this nonstandard setting and introduce vector-valued nonuniform multiresolution analysis (VNUMRA) where the associated subspace V0 of L2(K, ℂM) has an orthonormal basis of the form {Φ (x - λ)}λ∈Λ where Λ = {0, r/N} + 𝒵, N ≥ 1 is an integer and r is an odd integer such that r and N are relatively prime and 𝒵 = {u(n) : n ∈ ℕ0}. We establish a necessary and sufficient condition for the existence of associated wavelets and derive an algorithm for the construction of VNUMRA on local fields starting from a vector refinement mask G(ξ) with appropriate conditions. Further, these results also hold for Cantor and Vilenkin groups.

ON FRAME SYSTEMS IN BANACH SPACES

2009 ◽  
Vol 07 (01) ◽  
pp. 1-7 ◽  
Author(s):  
P. K. JAIN ◽  
S. K. KAUSHIK ◽  
NISHA GUPTA
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Bessel Sequence ◽  
Banach Frame ◽  
Frame System ◽  
Frame Systems ◽  
Necessary And Sufficient

Banach frame systems in Banach spaces have been defined and studied. A sufficient condition under which a Banach space, having a Banach frame, has a Banach frame system has been given. Also, it has been proved that a Banach space E is separable if and only if E* has a Banach frame ({φn},T) with each φn weak*-continuous. Finally, a necessary and sufficient condition for a Banach Bessel sequence to be a Banach frame has been given.

On fusion frames in Banach spaces

2010 ◽  
Vol 18 (1) ◽  
pp. 121-130
Author(s):  
Shiv K. Kaushik ◽  
Varinder Kumar
Keyword(s):  
Banach Spaces ◽  
Complete Sequence ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Banach Frames ◽  
Fusion Frames ◽  
Banach Frame ◽  
Necessary And Sufficient

Abstract A necessary and sufficient condition for a complete sequence of subspaces to be a fusion Banach frame for E is given. Also, we introduce fusion Banach frame sequences and give a characterization for a complete sequence of subspaces of E to be a fusion Banach frame for E in terms of fusion Banach frame sequences. Finally, along with other results, we characterize fusion Banach frames in terms of Banach frames.

scholarly journals A further necessary and sufficient condition for strong convergence of nonlinear contraction semigroups and of iterative methods for accretive operators in Banach spaces

1995 ◽  
Vol 38 (1) ◽  
pp. 1-12 ◽  
Author(s):  
Zong-Ben Xu ◽  
Yao-Lin Jiang ◽  
G. F. Roach
Keyword(s):  
Banach Space ◽  
Strong Convergence ◽  
Banach Spaces ◽  
Iterative Methods ◽  
Accretive Operator ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Accretive Operators ◽  
Uniformly Smooth ◽  
Necessary And Sufficient

Let A be a quasi-accretive operator defined in a uniformly smooth Banach space. We present a necessary and sufficient condition for the strong convergence of the semigroups generated by – A and of the steepest descent methods to a zero of A.

ON SOME CLASSES OF SCHAUDER FRAMES IN BANACH SPACES

2014 ◽  
Vol 12 (02) ◽  
pp. 1450022
Author(s):  
KHOLE TIMOTHY POUMAI
Keyword(s):  
Banach Spaces ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Necessary And Sufficient ◽  
Schauder Frame

Various types of Schauder frames have been defined and studied. A necessary and sufficient condition for each type of Schauder frame is given. Finally, we give some theoretical applications of these types of Schauder frames.

FRAMES OF SUBSPACES FOR BANACH SPACES

2010 ◽  
Vol 08 (02) ◽  
pp. 243-252 ◽  
Author(s):  
P. K. JAIN ◽  
S. K. KAUSHIK ◽  
VARINDER KUMAR
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Banach Frame ◽  
Necessary And Sufficient ◽  
Compactly Generated

Frames of subspaces for Banach spaces have been introduced and studied. Examples and counter-examples to distinguish various types of frames of subspaces have been given. It has been proved that if a Banach space has a Banach frame, then it also has a frame of subspaces. Also, a necessary and sufficient condition for a sequence of projections, associated with a frame of subspaces, to be unique has been given. Finally, we consider complete frame of subspaces and prove that every weakly compactly generated Banach space has a complete frame of subspaces.

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