scholarly journals The uniform boundedness theorem in b-Banach space

2021 ◽  
Author(s):  
Jiachen Lv ◽  
Yuqiang Feng
Keyword(s):  
Banach Space ◽  
Uniform Boundedness ◽  
Boundedness Theorem

scholarly journals Uniform boundedness principles for regular Borel vector measures

1980 ◽  
Vol 29 (2) ◽  
pp. 206-218 ◽  
Author(s):  
Joseph Kupka
Keyword(s):  
Banach Space ◽  
Borel Measure ◽  
Borel Subset ◽  
Uniform Boundedness ◽  
Vector Measures ◽  
Boundedness Theorem ◽  
Regular Borel Measure ◽  
Pointwise Limit ◽  
Regular Open ◽  
Uniformly Bounded

The setting is a compact Hausfroff space ω. The notion of a Walls class of subsets of Ω is defined via strange axioms—axioms whose justification rests with examples such as the collection of regular open sets or the range of a strong lifting. Avarient of Rosenthal' famous lwmma which applies directly to Banach space-valued measures is esablished, and it is used to obtain, in elementary fashion, the following two uniform boundedness principles: (1)The Nikodym Boundedness Theorem. If K is a family of regular Borel vector measures on Ω which is point-wise bounded on every set of a fixed Wells class, then K is uniformly bounded. (2)The Nikodym Covergence Theorem. If {μn} is a sequence of regular Borel vector measures on Ω which is converguent on every set of a fixed Wells class, then the μn are uniformly countably additive, the sequence {μn} is convergent on every Borel subset of Ω and the pointwise limit constitutes a regular Borel measure.

scholarly journals A uniform boundedness theorem

1967 ◽  
Vol 18 (4) ◽  
pp. 624-624
Author(s):  
John W. Brace ◽  
Robert M. Nielsen
Keyword(s):  
Uniform Boundedness ◽  
Boundedness Theorem

scholarly journals Bidual Spaces and Reflexivity of Real Normed Spaces

2014 ◽  
Vol 22 (4) ◽  
pp. 303-311
Author(s):  
Keiko Narita ◽  
Noboru Endou ◽  
Yasunari Shidama
Keyword(s):  
Real Number ◽  
Normed Space ◽  
Uniform Boundedness ◽  
Linear Operators ◽  
Linear Functionals ◽  
Normed Spaces ◽  
Boundedness Theorem ◽  
Natural Mapping ◽  
Real Normed Space ◽  
Banach Theorem

Summary In this article, we considered bidual spaces and reflexivity of real normed spaces. At first we proved some corollaries applying Hahn-Banach theorem and showed related theorems. In the second section, we proved the norm of dual spaces and defined the natural mapping, from real normed spaces to bidual spaces. We also proved some properties of this mapping. Next, we defined real normed space of R, real number spaces as real normed spaces and proved related theorems. We can regard linear functionals as linear operators by this definition. Accordingly we proved Uniform Boundedness Theorem for linear functionals using the theorem (5) from [21]. Finally, we defined reflexivity of real normed spaces and proved some theorems about isomorphism of linear operators. Using them, we proved some properties about reflexivity. These formalizations are based on [19], [20], [8] and [1].

scholarly journals UNCONDITIONAL CONVERGENCE IN THE STRONG OPERATOR TOPOLOGY AND ℓ∞

2011 ◽  
Vol 53 (3) ◽  
pp. 583-598 ◽  
Author(s):  
IOANA GHENCIU ◽  
PAUL LEWIS
Keyword(s):  
Banach Space ◽  
Vector Measure ◽  
Strong Operator Topology ◽  
Unconditional Convergence ◽  
Fundamental Result ◽  
Boundedness Theorem ◽  
Strong Operator ◽  
Spaces Of Operators ◽  
Funct Anal ◽  
Nikodym Theorem

AbstractIn this paper we study non-complemented spaces of operators and the embeddability of ℓ∞ in the spaces of operators L(X, Y), K(X, Y) and Kw*(X*, Y). Results of Bator and Lewis [2, 3] (Bull. Pol. Acad. Sci. Math.50(4) (2002), 413–416; Bull. Pol. Acad. Sci. Math.549(1) (2006), 63–73), Emmanuele [8–10] (J. Funct. Anal.99 (1991), 125–130; Math. Proc. Camb. Phil. Soc.111 (1992), 331–335; Atti. Sem. Mat. Fis. Univ. Modena42(1) (1994), 123–133), Feder [11] (Canad. Math. Bull.25 (1982), 78–81) and Kalton [16] (Math. Ann.208 (1974), 267–278), are generalised. A vector measure result is used to study the complementation of the spaces W(X, Y) and K(X, Y) in the space L(X, Y), as well as the complementation of K(X, Y) in W(X, Y). A fundamental result of Drewnowski [7] (Math. Proc. Camb. Phil. Soc. 108 (1990), 523–526) is used to establish a result for operator-valued measures, from which we obtain as corollaries the Vitali–Hahn–Saks–Nikodym theorem, the Nikodym Boundedness theorem and a Banach space version of the Phillips Lemma.

scholarly journals The Holomorphic Hörmander Functional Calculus

2019 ◽  
Author(s):  
Florian Pannasch
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Functional Calculus ◽  
Differential Operators ◽  
Uniform Boundedness ◽  
General Context ◽  
Multiplier Theorem ◽  
Integral Means ◽  
Imaginary Powers ◽  
Multiplier Theorems

TThe topic of this thesis is functional calculus in connection with abstract multiplier theorems. In 1960, Hörmander showed how the uniform boundedness of certain integral means of a function m in L ∞ (R^d) and its weak derivatives imply that m yields a bounded Lp -Fourier multiplier. Nowadays, this is known as the Hörmander multiplier theorem, sometimes Hörmander--Mikhlin multiplier theorem. A noteworthy detail is that a radial function m(|x|) satisfies Hörmander's condition if and only if m (|x|²) does. Hence, Hörmander's theorem is also a result on the functional calculus of the negative Laplacian -Δ. Hörmander's result has inspired a lot of research, and authors have also proven similar results for other operators such as certain Schrödinger operators, Sublaplacians on Lie groups, and later certain differential operators on spaces of homogeneous type. For us, the work of Kriegler and Weis is of particular interest. Starting with the PhD thesis of Kriegler in 2009, they showed how abstract multiplier theorems can be proven in a more general context. Namely, considering a certain class of 0-sectorial and 0-strip type operators on a general Banach space, one can construct an abstract Hörmander functional calculus based on the classical holomorphic calculus. Then, by using probalistic techniques from Banach space geometry involving so-called R-boundedness one can derive multiplier results in this generalized setting. In 2001, García-Cuerva, Mauceri, Meda, Sjögren, and Torrea proved an abstract multiplier theorem for generators of symmetric contraction semigroups, where a bounded Hörmander calculus is inferred from growth conditions on the imaginary powers of the generator. As the considered operators need not be 0-sectorial, this result is not covered by the methods of Kriegler and Weis. However, the result is based on Meda's earlier work, where he derived a bounded Hörmander if the given imaginary powers only grow polynomially fast. In this case, the operator is 0-sectorial, and Kriegler and Weis were able to recover the result while improving the order of the calculus. In this thesis, we introduce a generalized class of Hörmander functions defined on strips and sectors. Based on this and the classical holomorphic calculus, we construct a holomorphic Hörmander calculus for a class of operators which may also have strip type or angle of sectoriality greater than zero. The main result is a generalization of the multiplier theorem of García-Cuerva et al. to Banach spaces of finite cotype and Banach spaces with Pisier's property (α), where we retain and even improve the order given by Kriegler and Weis for the 0-sectorial case.

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