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Dissertations / Theses on the topic 'Banach space'
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1
Albasrawi, Fatimah Hassan. "Floquet Theory on Banach Space." TopSCHOLAR®, 2013. http://digitalcommons.wku.edu/theses/1234.
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In this thesis we study Floquet theory on a Banach space. We are concerned about the linear differential equation of the form: y'(t) = A(t)y(t), where t ∈ R, y(t) is a function with values in a Banach space X, and A(t) are linear, bounded operators on X. If the system is periodic, meaning A(t+ω) = A(t) for some period ω, then it is called a Floquet system. We will investigate the existence and uniqueness of the periodic solution, as well as the stability of a Floquet system. This thesis will be presented in five main chapters. In the first chapter, we review the system of linear differential equations on Rn: y'= A(t)y(t) + f(t), where A(t) is an n x n matrix-valued function, y(t) are vectors and f(t) are functions with values in Rn. We establish the general form of the all solutions by using the fundamental matrix, consisting of n independent solutions. Also, we can find the stability of solutions directly by using the eigenvalues of A. In the second chapter, we study the Floquet system on Rn, where A(t+ω) = A(t). We establish the Floquet theorem, in which we show that the Floquet system is closely related to a linear system with constant coefficients, so the properties of those systems can be applied. In particular, we can answer the questions about the stability of the Floquet system. Then we generalize the Floquet theory to a linear system on Banach spaces. So we introduce to the readers Banach spaces in chapter three and the linear operators on Banach spaces in chapter four. In the fifth chapter we study the asymptotic properties of solutions of the system: y'(t) = A(t)y(t), where y(t) is a function with values in a Banach space X and A(t) are linear, bounded operators on X with A (t+ω) = A(t). For that system, we can show the evolution family U(t,s) representing the solutions is periodic, i.e. U(t+ω, s+ω) = U(t,s). Next we study the monodromy of the system V := U(ω,0). We point out that the spectrum set of V actually determines the stability of the Floquet system. Moreover, we show that the existence and uniqueness of the periodic solution of the nonhomogeneous equation in a Floquet system is equivalent to the fact that 1 belongs to the resolvent set of V.
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Jiang, Zhu. "Topics in Banach space theory." Thesis, Lancaster University, 1991. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.316579.
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Boedihardjo, March Tian. "Topics in Banach space theory." HKBU Institutional Repository, 2011. http://repository.hkbu.edu.hk/etd_ra/1286.
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Das, Bata Krishna. "Quantum stochastic analysis in Banach space and operator space." Thesis, Lancaster University, 2012. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.660115.
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Derrick, John. "Some problems in Banach space theory." Thesis, University of Oxford, 1988. http://ora.ox.ac.uk/objects/uuid:36289504-6d9f-42e4-af95-ef3abb8a8fa2.
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Types were introduced by Krivine and Maurey, in a refinement of a result by Aldous showing that infinite dimensional subspaces of Lr contain Ωp for some 1≤pꝏ) . A synthesis of these ideas was provided by Garling whose representation of types as random measures was the motivation for much of this work. This thesis aims to investigate the structure of the representation, and to provide concrete representations for differing Banach spaces. Chapter one contains the necessary preliminaries for the later chapters, and finishes by introducing the representation due to Garling of types on Lϕ(X) as random measures on τ(X) The second chapter consists of two parts. In the first part we examine the structure of the map between types on Lp(X) and random measures on τ(X) . We show that convolution is preserved by the mapping, and give an explicit representation of the space of types on L1(Ωp). The second part is concerned with representations of τ(X) . We give conditions for the decomposition of τ(X) into X*S(X) , and derive representations for the space of types on L1(L2k). The third chapter studies differentiability of types. We extend differentiability from X to τ(X) , and develop ideas that will be used in the study of uniqueness. In chapter four we consider questions concerning the uniqueness of measures and random measures on X and τ(X) . We construct spaces where the representation of types as random measures is not in uniquely determined. We prove that if a certain uniqueness property for measures on X fails then Ωn1 embeds in X.
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Vuong, Thi Minh Thu. "Complemented and uncomplemented subspaces of Banach spaces." Thesis, University of Ballarat, 2006. http://researchonline.federation.edu.au/vital/access/HandleResolver/1959.17/51906.
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"A natural process in examining properties of Banach spaces is to see if a Banach space can be decomposed into simpler Banach spaces; in other words, to see if a Banach space has complemented subspaces. This thesis concentrates on three main aspects of this problem: norm of projections of a Banach space onto its finite dimensional subspaces; a class of Banach spaces, each of which has a large number of infinite dimensional complemented subspaces; and methods of finding Banach spaces which have uncomplemented subspaces, where the subspaces and the quotient spaces are chosen as well-known classical sequence spaces (finding non-trivial twisted sums)." --Abstract.Master of Mathematical Sciences
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Vuong, Thi Minh Thu. "Complemented and uncomplemented subspaces of Banach spaces." University of Ballarat, 2006. http://archimedes.ballarat.edu.au:8080/vital/access/HandleResolver/1959.17/15540.
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"A natural process in examining properties of Banach spaces is to see if a Banach space can be decomposed into simpler Banach spaces; in other words, to see if a Banach space has complemented subspaces. This thesis concentrates on three main aspects of this problem: norm of projections of a Banach space onto its finite dimensional subspaces; a class of Banach spaces, each of which has a large number of infinite dimensional complemented subspaces; and methods of finding Banach spaces which have uncomplemented subspaces, where the subspaces and the quotient spaces are chosen as well-known classical sequence spaces (finding non-trivial twisted sums)." --Abstract.Master of Mathematical Sciences
8
Kiteu, Marco M. "Orbits of operators on Hilbert space and some classes of Banach spaces." Kent State University / OhioLINK, 2012. http://rave.ohiolink.edu/etdc/view?acc_num=kent1341850621.
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Koller, Michael Dominik Fabian. "Topologies on the set of Banach space representations /." [S.l.] : [s.n.], 1993. http://e-collection.ethbib.ethz.ch/show?type=diss&nr=10075.
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Terauds, Venta School of Mathematics UNSW. "Extentions of functional calculus for Banach space operators." Awarded by:University of New South Wales. School of Mathematics, 2006. http://handle.unsw.edu.au/1959.4/25726.
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We consider conditions under which a continuous functional calculus for a Banach space operator T ?? L(X) may be extended to a bounded Borel functional calculus, and under which a functional calculus for absolutely continuous (AC) functions may be extended to one of for functions of bounded variation (BV). The natural setting for investigating the former case is finitely spectral operators, and for the latter, well-bounded operators. Some such conditions are well-established. If X is a reflexive space, both type of Extensions are assured; in fact if X contains an isomorphic copy of co, then every Operator T ?? L(X) that has a continuous functional calculus necessarily admits a Borel one. We show that if a space X has a predual, then also every operator T ?? L(X) with a continuous functional calculus admits a bounded Borel functional Calculus. In case a Banach space X either contains an isomorphic copy of co, or has a Predual, and T ?? L(X) is an operator with an AC functional calculus, we find that the existence of a decomposition of the identity of bounded variation for T is sufficient to ensure that the AC functional calculus may be extended to a BV functional calculus. We also consider operators defined by a linear map on interpolation families of Banach spaces [Xr, X???] (r???1), where for example Xp = lp, Lp[0,1] or Cp. We show that under certain uniform boundedness conditions, the possession of a BV functional calculus by operators on the spaces Xp, p ?? (r, ???), may be extrapolated to the corresponding operators on the spaces Xr and X???.
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RUSSO, TOMMASO. "ON SOME OPEN PROBLEMS IN BANACH SPACE THEORY." Doctoral thesis, Università degli Studi di Milano, 2019. http://hdl.handle.net/2434/609289.
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The main line of investigation of the present work is the study of some aspects in the analysis
of the structure of the unit ball of (infinite-dimensional) Banach spaces. In particular,
we analyse some questions concerning the existence of suitable renormings that allow the
new unit ball to possess a specific geometric property. The main part of the thesis is,
however, dedicated to results of isometric nature, in which the original norm is the one
under consideration.
One of the main sources for the selection of the topics of investigation has been the
recent monograph [GMZ16], entirely dedicated to collecting several open problems in
Banach space theory and formulating new lines of investigation. We take this opportunity
to acknowledge the authors for their effort, that offered such useful text to the
mathematical community. The results to be discussed in our work actually succeed in
solving a few of the problems presented in the monograph and are based on the papers
[HáRu17, HKR18, HáRu19, HKR••].
Let us say now a few words on how the material is organised. The thesis is divided
in four chapters (some whose contents are outlined below) which are essentially independent
and can be read in whatsoever order. The unique chapter which is not completely
independent from the others is Chapter 4, where we use some results from Chapter 2 and
which is, in a sense, the non-separable prosecution of Chapter 3. However, cross-references
are few (never implicit) and usually restricted to quoting some result; it should therefore
be no problem to start reading from Chapter 4.
The single chapters all share the same arrangement. A first section is dedicated to an
introduction to the subject of the chapter; occasionally, we also present the proof of known
results, in most cases as an illustration of an important technique in the area. In these
introductions we strove to be as self-contained as possible in order to help the novel reader
to enter the field; consequently, experts in the area may find them somewhat redundant
and prefer to skip most parts of them.
The first section of each chapter concludes with the statement of our most significant
results and a comparison with the literature. The proofs of these results, together with
additional results or generalisations, are presented in the remaining sections of the chapter.
These sections usually follow closely the corresponding articles (carefully referenced) where
the results were presented.
12
VISCARDI, MARIA CLAUDIA. "ALMOST TRANSITIVE AND ALMOST HOMOGENEOUS SEPARABLE BANACH SPACES." Doctoral thesis, Università degli Studi di Milano, 2018. http://hdl.handle.net/2434/582009.
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In this thesis we provide an overview of a problem related to almost homogeneous
separable Banach spaces; in particular we focus on a construction of the Gurarii
space due to J. Garbuli«ska and W. Kubis and some of its consequences,
giving a personal contribution to the state of the art. In particular we focused on the amalgamation property of the class of all finite-dimensional Banach spaces.
13
Hofmann, Bernd, and Peter Mathé. "Parameter choice in Banach space regularization under variational inequalities." Universitätsbibliothek Chemnitz, 2012. http://nbn-resolving.de/urn:nbn:de:bsz:ch1-qucosa-86241.
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The authors study parameter choice strategies for Tikhonov regularization of nonlinear ill-posed problems in Banach spaces. The effectiveness of any parameter choice for obtaining convergence rates depend on the interplay of the solution smoothness and the nonlinearity structure, and it can be expressed concisely in terms of variational inequalities. Such inequalities are link conditions between the penalty term, the norm misfit and the corresponding error measure. The parameter choices under consideration include an a priori choice, the discrepancy principle as well as the Lepskii principle. For the convenience of the reader the authors review in an appendix a few instances where the validity of a variational inequality can be established.
14
Athapattu, Chathurika Umayangani. "PARABOLICALLY INDUCED BANACH SPACE REPRESENTATION OF P-ADIC GROUPS." OpenSIUC, 2020. https://opensiuc.lib.siu.edu/dissertations/1783.
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Parabolic induction is a method of constructing representations of a reductive group from representations of its parabolic subgroups. We study the parabolic induction for p-adic Banach space representation of p-adic groups. In this dissertation, using the Schneider-Teitelbaum duality, we consider the corresponding Iwasawa modules. We present the dual of parabolically induced representation in terms of the tensor product.
15
Malý, Lukáš. "Newtonian Spaces Based on Quasi-Banach Function Lattices." Licentiate thesis, Linköpings universitet, Matematik och tillämpad matematik, 2012. http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-79166.
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The traditional first-order analysis in Euclidean spaces relies on the Sobolev spaces W1,p(Ω), where Ω ⊂ Rn is open and p ∈ [1, ∞].The Sobolev norm is then defined as the sum of Lp norms of a function and its distributional gradient.We generalize the notion of Sobolev spaces in two different ways. First, the underlying function norm will be replaced by the “norm” of a quasi-Banach function lattice. Second, we will investigate functions defined on an abstract metric measure space and that is why the distributional gradients need to be substituted. The thesis consists of two papers. The first one builds up the elementary theory of Newtonian spaces based on quasi-Banach function lattices. These lattices are complete linear spaces of measurable functions with a topology given by a quasinorm satisfying the lattice property. Newtonian spaces are first-order Sobolev-type spaces on abstract metric measure spaces, where the role of weak derivatives is passed on to upper gradients. Tools such asmoduli of curve families and the Sobolev capacity are developed, which allows us to study basic properties of the Newtonian functions.We will see that Newtonian spaces can be equivalently defined using the notion of weak upper gradients, which increases the number of techniques available to study these spaces. The absolute continuity of Newtonian functions along curves and the completeness of Newtonian spaces in this general setting are also established. The second paper in the thesis then continues with investigation of properties of Newtonian spaces based on quasi-Banach function lattices. The set of all weak upper gradients of a Newtonian function is of particular interest.We will prove that minimalweak upper gradients exist in this general setting.Assuming that Lebesgue’s differentiation theoremholds for the underlyingmetricmeasure space,wewill find a family of representation formulae. Furthermore, the connection between pointwise convergence of a sequence of Newtonian functions and its convergence in norm is studied.
16
Brackebusch, Ruth Elaine. "James space on general trees /." The Ohio State University, 1985. http://rave.ohiolink.edu/etdc/view?acc_num=osu1487260135357124.
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Yavuz, Onur. "Invariant subspaces for Banach space operators with a multiply connected spectrum." [Bloomington, Ind.] : Indiana University, 2006. http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqdiss&rft_dat=xri:pqdiss:3219888.
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Thesis (Ph.D.)--Indiana University, Dept. of Mathematics, 2006."Title from dissertation home page (viewed June 27, 2007)." Source: Dissertation Abstracts International, Volume: 67-06, Section: B, page: 3174. Adviser: Hari Bercovici.
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Wimelaratna, Ramasinghege. "Multi dimensional geometric moduli and exterior algebra of a Banach space /." The Ohio State University, 1988. http://rave.ohiolink.edu/etdc/view?acc_num=osu148759830383865.
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D'Alessandro, S. "POLYNOMIAL ALGEBRAS AND SMOOTH FUNCTIONS IN BANACH SPACES." Doctoral thesis, Università degli Studi di Milano, 2014. http://hdl.handle.net/2434/244407.
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According to the fundamental Stone-Weierstrass theorem, if X is a finite dimensional real Banach space, then every continuous function on the unit ball B_X can be uniformly approximated by polynomials. For infinite dimensional Banach spaces the statement of the Stone-Weierstrass Theorem is false, even if we replace continuous functions by the uniformly continuous ones (which is a natural condition that coincides with continuity in the finite dimensional setting): in fact, on every infinite-dimensional Banach space X there exists a uniformly continuous real function not approximable by continuous polynomials.
The natural problem of the proper generalization of the result for infinite dimensional spaces was posed by Shilov (in the case of a Hilbert space). Aron observed that the uniform closure on B_X of the space of all polynomials of the finite type is precisely the space of all functions which are weakly uniformly continuous on B_X. Since there exist infinite dimensional Banach spaces such that all bounded polynomials are weakly uniformly continuous on B_X (e.g. C_0 or more generally all Banach spaces not containing a copy of l_1 and such that all bounded polynomials are weakly sequentially continuous on B_X), this result gives a very satisfactory solution to the problem. Unfortunately, most Banach spaces, including L_p, do not have this special property. In this case, no characterization of the uniform limits of polynomials is known. But the problem has a more subtle formulation as well. Let us consider the algebras consisting of all polynomials which can be generated by finitely many algebraic operations of addition and multiplication, starting from polynomials on X of degree not exceeding n. Of course, such polynomials can have arbitrarily high degree. It is clear that, if n is the lowest degree such that there exists a polynomial P which is not weakly uniformly continuous, then the we have equalities among the algebras up to n-1 and then we have a strict inclusion. The problem of what happens from n on has been studied in several papers. The natural conjecture appears to be that once the chain of eualities has been broken, it is going to be broken at each subsequent step. The proof of this latter statement given by Hajek in 1996, for all classical Banach spaces, based on the theory of algebraic bases, is unfortunately not entirely correct, as was pointed out by our colleague Michal Johanis. It is not clear to us if the theory of algebraic bases developed therein can be salvaged. Fortunately, the main statement of this theory can be proved using another approach. The complete proof can be found in this thesis. Most of the results in this area are therefore safe.
The main result of this thesis implies all previously known results in this area (all confirming the above conjecture) as special cases.
We also give solutions to three other problems posed in the literature, which are concerning smooth functions rather than polynomials, but which belong to the same field of study of smooth mappings on a Banach space.
The first result is a construction of a non-equivalent C^k-smooth norm on every Banach space admitting a C^k-smooth norm, answering a problem posed in several places in the literature.
We solve a another question by proving that a real Banach space admitting a separating real analytic function whose holomorphic extension is Lipschitz in some strip around X admits a separating polynomial.
Eventually, we solve a problem posed by Benyamini and Lindenstrauss, concerning the extensions of uniformly differentiable functions from the unit ball into a larger set, preserving the values in some neighbourhood of the origin. More precisely, we construct an example of a uniformly differentiable real-valued function f on the unit ball of a certain Banach space X, such that there exists no uniformly differentiable function g on cB_X for any c>1 which coincides with f in some neighbourhood of the origin. To do so, we construct suitable renormings of c_0, based on the theory of W-spaces.
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Garrisi, Daniele. "Ordinary differential equations in Banach spaces and the spectral flow." Doctoral thesis, Scuola Normale Superiore, 2008. http://hdl.handle.net/11384/85668.
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Ronquillo, Rivera Javier Alfredo. "Extremely Amenable Groups and Banach Representations." Ohio University / OhioLINK, 2018. http://rave.ohiolink.edu/etdc/view?acc_num=ohiou1520548085599864.
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Carrera, Wilson Albeiro Cuellar. "Espaços de Banach com várias estruturas complexas." Universidade de São Paulo, 2015. http://www.teses.usp.br/teses/disponiveis/45/45131/tde-04092016-203116/.
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No presente trabalho, estudamos alguns aspectos da teoria de estruturas complexas em espaços de Banach. Demonstramos que se um espaço de Banach real $X$ tem a propriedade $P$, então todas as estruturas complexas em $X$ também satisfazem $P$, quando $P$ é qualquer uma das seguintes propriedades: propriedade de aproximação limitada, \\emph{G.L-l.u.st}, ser injetivo e ser complementado num espaço dual. Abordamos o problema da unicidade de estruturas complexas em espaços de Banach com base subsimétrica, provando que um espaço de Banach real $E$ com base subsimétrica e isomorfo ao espaço de sequências $E[E]$ admite estrutura complexa única. Por outro lado, apresentamos um exemplo de espaço de Banach com exatamente $\\omega$ estruturas complexas distintas. Também usamos a teoria de estruturas complexas para estudar o clássico problema dos hiperplanos no espaço $Z_2$ de Kalton-Peck. Com o propósito de distinguir $Z_2$ de seus hiperplanos nos perguntamos se os hiperplanos admitem estrutura complexa. Nesse sentido, provamos que os hiperplanos de $Z_2$ contendo a cópia canônica de $\\ell_2$ não admitem estruturas complexas que sejam extensões de estruturas complexas em $\\ell_2$. Também construímos uma estrutura complexa em $\\ell_2$ que não pode-se estender a nenhum operador em $Z_2$.In this work, we study some aspects of the theory of complex structures in Banach spaces. We show that if a real Banach space $X$ has the property $P$, then all its complex structures also satisfy $P$, where $P$ is any of the following properties: bounded approximation property, \\emph{G.L-l.u.st}, being injective and being complemented in a dual space. We address the problem of uniqueness of complex structures in Banach spaces with subsymmetric basis by proving that a real Banach space $E$ with subsymmetric basis and isomorphic to the space of sequences $E [E]$ admits a unique complex structure. On the other hand, we show an example of Banach space with exactly $\\omega$ different complex structures. We also use the theory of complex structures to study the classical problem of hyperplanes in the Kalton-Peck space $Z_2$. In order to distinguish between $Z_2$ and its hyperplanes we wonder whether the hyperplanes admit complex structures. In this sense we prove that no complex structure on $\\ell_2$ can be extended to a complex structure on the hyperplanes of $Z_2$ containing the canonical copy $l_2$. We also constructed a complex structure on $l_2$ that can not be extended to any operator in $Z_2$.
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Azimi, Mahdi [Verfasser]. "Banach space valued stochastic integral equations and their optimal control / Mahdi Azimi." Halle, 2018. http://d-nb.info/1153401991/34.
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Farmer, Matthew Ray. "Applications in Fixed Point Theory." Thesis, University of North Texas, 2005. https://digital.library.unt.edu/ark:/67531/metadc4971/.
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Banach's contraction principle is probably one of the most important theorems in fixed point theory. It has been used to develop much of the rest of fixed point theory. Another key result in the field is a theorem due to Browder, Göhde, and Kirk involving Hilbert spaces and nonexpansive mappings. Several applications of Banach's contraction principle are made. Some of these applications involve obtaining new metrics on a space, forcing a continuous map to have a fixed point, and using conditions on the boundary of a closed ball in a Banach space to obtain a fixed point. Finally, a development of the theorem due to Browder et al. is given with Hilbert spaces replaced by uniformly convex Banach spaces.
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Yeates, Stephen. "Non-quasianalytic representations of semigroups : their spectra and asymptotics." Thesis, University of Oxford, 1998. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.297366.
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Erkursun, Nazife. "Convergence Of Lotz-raebiger Nets On Banach Spaces." Phd thesis, METU, 2010. http://etd.lib.metu.edu.tr/upload/3/12612108/index.pdf.
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The concept of LR-nets was introduced and investigated firstly by H.P. Lotz in [27] and by F. Raebiger in [30]. Therefore we call such nets Lotz-Raebiger nets, shortly LR-nets. In this thesis
we treat two problems on asymptotic behavior of these operator nets.
First problem is to generalize well known theorems for Ces`aro averages of a single operator to LR-nets, namely to generalize the Eberlein and Sine theorems. The second problem is related
to the strong convergence of Markov LR-nets on L1-spaces. We prove that the existence of a lower-bound functions is necessary and sufficient for asymptotic stability of LR-nets of
Markov operators.
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Baratov, Rishat. "Efficient conic decomposition and projection onto a cone in a Banach ordered space." Thesis, University of Ballarat, 2005. http://researchonline.federation.edu.au/vital/access/HandleResolver/1959.17/61401.
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Lian, Zeng. "Lyapunov Exponents and Invariant Manifold for Random Dynamical Systems in a Banach Space." Diss., CLICK HERE for online access, 2008. http://contentdm.lib.byu.edu/ETD/image/etd2555.pdf.
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Sunehag, Peter. "Interpolation of Subcouples, New Results and Applications." Doctoral thesis, Uppsala : Matematiska institutionen, Univ. [distributör], 2003. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-3777.
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Crewe, Paul. "Some problems in abstract stochastic differential equations on Banach spaces." Thesis, University of Oxford, 2011. http://ora.ox.ac.uk/objects/uuid:195c374f-3181-41b9-92d7-b375701a0b81.
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This thesis studies abstract stochastic differential equations on Banach spaces. The well-posedness of abstract stochastic differential equations on such spaces is a recent result of van Neerven, Veraar and Weis, based on the theory of stochastic integration of Banach space valued processes constructed by the same authors. We study existence and uniqueness for solutions of stochastic differential equations with (possibly infinite) delay in their inputs on UMD Banach spaces. Such problems are also known as functional differential equations or delay differential equations. We show that the methods of van Neerven et al. extend to such problems if the initial history of the system lies in a space of a type introduced by Hale and Kato. The results are essentially of a fixed point type, both autonomous and non-autonomous cases are discussed and an example is given. We also study some long time properties of solutions to these stochastic differential equations on general Banach spaces. We show the existence of solutions to stochastic problems with almost periodicity in a weak or distributional sense. Results are again given for both autonomous and non-autonomous cases and depend heavily on estimates for R-bounds of operator families developed by Veraar. An example is given for a second order differential operator on a domain in ℝd. Finally we consider the existence of invariant measures for such problems. This extends recent work of van Gaans in Hilbert spaces to Banach spaces of type 2.
31
Bjorkman, Kaitlin. "A Weak Groethendieck Compactness Principle for Infinite Dimensional Banach Spaces." VCU Scholars Compass, 2013. http://scholarscompass.vcu.edu/etd/3042.
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The goal of this thesis is to give an exposition of the following recent result of Freeman, Lennard, Odell, Turett and Randrianantoanina. A Banach space has the Schur property if and only if every weakly compact set is contained in the closed convex hull of a weakly null sequence. This result complements an old result of Grothendieck (now called the Grothendieck Compactness Principle) stating that every norm compact subset of a Banach space is contained in the closed convex hull of a norm null sequence. We include many of the relevant definitions and preliminary results which are required in the proofs of both of these theorems.
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Rodríguez, Ruiz José. "Integración en espacios de Banach." Doctoral thesis, Universidad de Murcia, 2006. http://hdl.handle.net/10803/10963.
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Esta tesis doctoral se enmarca dentro de la teoría de integración de funciones con valores en espacios de Banach. Analizamos con detalle la integral de Birkhoff de funciones vectoriales, así como sus correspondientes versiones dentro de los contextos de la integración respecto de medidas vectoriales y la integración de multi-funciones. Comparamos estos métodos de integración con otros bien conocidos (integrales de Bochner, Pettis, McShane, Debreu, etc.). Caracterizamos, en términos de integración vectorial, algunas propiedades de los espacios de Banach donde las (multi-) funciones toman valores.The general framework of this memoir is the theory of integration of functions with values in Banach spaces. We analyze in detail the Birkhoff integral of vector-valued functions, as well as its corresponding versions within the settings of integration with respect to vector measures and integration of multi-valued functions. We compare these methods of integration with others which are well known (Bochner, Pettis, McShane, Debreu, etc.). We characterize, in terms of vector integration, some properties of the Banach spaces where the (multi-) functions take their values.
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Monteiro, Giselle Antunes. "Generalized linear differential equations in a Banach space: continuous dependence on parameters and applications." Universidade de São Paulo, 2012. http://www.teses.usp.br/teses/disponiveis/55/55135/tde-30032012-105214/.
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The purpose of this work is to investigate continuous dependence on parameters for generalized linear differential equations in a Banach space- valued setting. More precisely, we establish a theorem inspired by the clas- sical continuous dependence result due to Z. Opial. In addition, our second outcome extends, to Banach spaces, the result proved by M. Ashordia in the framework of finite dimensional generalized linear differential equations. Roughly speaking, the continuous dependence derives from assumptions of uniform convergence of the functions in the right-hand side of the equations, together with the uniform boundedness of variation of the linear terms. Fur- thermore, applications of these results to dynamic equations on time scales and also to functional differential equations are proposed. Besides these results on continuous dependence, we complete the theory of abstract Kurzweil-Stieltjes integration so that it is well applicable for our purposes in generalized linear differential equations. In view of this, our contributions are related not only to differential equations but also to the abstract Kurzweil-Stieltjes integration theory itself. The new results presented in this work are contained in the papers [26] and [27], both accepted for publicationO objetivo deste trabalho é investigar a dependência contínua de soluções em relação a parâmetros para equações diferenciais lineares generalizadas no contexto de espaços de Banach. Mais precisamente, apresentamos um teo- rema inspirado no resultado clássico de dependência contínua obtido por Z. Opial. Nosso segundo resultado estende, para espaços de Banach, o provado por M. Ashordia no contexto de equações diferenciais lineares gen- eralizadas em dimensão finita. Em linhas gerais, a dependência contínua decorre da convergência uniforme das funções à direita das equações, junta- mente com a limitação uniforme da variação dos termos lineares. No mais, são propostas aplicações desses resultados em equações dinâmicas em escalas temporais e também em equações diferenciais funcionais. Além dos resultados em dependência contínua, completamos à teoria de integração abstrata de Kurzweil-Stieltjes de modo que esta se adeque aos nossos propósitos em equações diferenciais lineares generalizadas. Assim, nossas contribuições dizem respeito não apenas a equações diferenciais, mas também a teoria de integração abstrata de Kurzweil-Stieltjes em si. Os resultados originais apresentados neste trabalho estão contidos nos artigos [26] e [27], ambos aceitos para publicação
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Portal, Pierre. "Harmonic analysis of banach space valued functions in the study of parabolic evolution equations /." free to MU campus, to others for purchase, 2004. http://wwwlib.umi.com/cr/mo/fullcit?p3137737.
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Abdellaoui, Taoufiq. "Distances de deux lois dans les espaces de Banach." Rouen, 1994. http://www.theses.fr/1994ROUE5003.
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La distance entre deux lois est étudiée lorsque les probabilités sont définies sur un espace de Banach séparable. Nous montrons que cette distance est atteinte par une fonction, dite associée, lorsque l'une des lois est diffuse l'autre discrète. Une condition nécessaire et suffisante pour reconnaitre le caractère associé est donnée par la cyclique monotonie. De plus un algorithme est donné pour obtenir de manière effective la fonction associée. Lorsque nous sommes dans un Hilbert séparable ces résultats sont étendus en utilisant des techniques de sous-gradients et d'analyse convexe
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Kuo, Po Ling. "Operadores de extensão de aplicações multilineares ou polinomios homogeneos." [s.n.], 2007. http://repositorio.unicamp.br/jspui/handle/REPOSIP/307329.
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Orientador: Jorge Tulio Mujica AscuiTese (doutorado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica
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Resumo: Este trabalho está dedicado ao estudo dos operadores de Nicodemi, introduzidos em [7] a partir de uma idéia em [12]. Os operadores de Nicodemi levam aplicações multilineares (resp. polinômios homogêneos) de um espaço de Banach E em aplicações multilineares (resp. polinômios homogêneos) em um espaço de Banach F. O nosso primeiro objetivo é encontrar condições para que os operadores de Nicodemi preservem certos tipos de aplicações multilineares (resp. polinômios homogêneos). Em particular estudamos a preservação de aplicações multilineares simétricas, de tipo finito, nucleares, compactas ou fracamente compactas. O segundo objetivo é encontrar condições para que, se os espaços duais E¿ e F¿ são isomorfos, os espaços de aplicações multilineares (resp. polinômios homogêneos) em E e F sejam isomorfos também. Estudamos também o problema correspondente para os espaços de aplicações multilineares (resp. polinômios homogêneos) de um determinado tipo, como por exemplo, de tipo finito, nuclear, compacto ou fracamente compacto
Abstract: This work is devoted to studying the Nicodemi operators, introduced in [7], following an idea in [12]. The Nicodemi operators map multilinear mappings (resp. homogeneous polynomials) on a Banach spaces E into multilinear mappings (resp. homogeneous polynomials) on a Banach spaces F. Our first objective is to find conditions under which the Nicodemi operators preserve certain types of multilinear mappings (resp. homogeneous polynomials). In particular we examine the preservation of the multilinear mappings that are symmetric, of finite type, nuclear, compact or weakly compact. Our second objective is tofind conditions under which, whenever the dual spaces E¿ and F¿ are isomorphic, the spaces of multilinear mappings (resp. homogeneous polynomials) on E and F are isomorphic as well. We also examine the corresponding problem for the spaces of multilinear mappings (resp. homogeneous polynomials) of a certain type, for instance of finite, nuclear, compact or weakly compact type
Doutorado
Analise Funcional
Doutor em Matemática
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Barbeiro, André Santoleri Villa. "Extensões conexas e espaços de Banach C(K) com poucos operadores." Universidade de São Paulo, 2018. http://www.teses.usp.br/teses/disponiveis/45/45131/tde-19042018-123305/.
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Este trabalho tem dois objetivos principais. Primeiramente, analisamos a preservação de conexidade na extensão de espaços compactos por funções contínuas, técnica utilizada por Koszmider para obter $C(K)$ indecomponível com poucos operadores. Mostramos que para todo compacto metrizável $K$ existe um desconexo $L$ que é obtido a partir de $K$ por uma quantidade finita de extensões por funções contínuas. Em seguida, enfatizamos a construção de espaços de Banach da forma $C(K)$ com poucos operadores, com a propriedade de que $C(L)$ tem poucos operadores, para todo fechado $L \\subseteq K$. Assumindo o princípio diamante construímos uma família $(K_\\xi)_{\\xi < 2^{(2^\\omega)}}$ de espaços conexos e hereditariamente Koszmider tais que todo operador de $C(K_\\xi)$ em $C(K_\\eta)$ é fracamente compacto, para $\\xi$ diferente de $\\eta$. Em particular, $(C(K_\\xi))_{\\xi < 2^{(2^\\omega)}}$ é uma família de espaços de Banach indecomponíveis e dois a dois essencialmente incomparáveis, e cada espaço $K_\\xi$ responde positivamente ao problema de Efimov. Apresentamos também um método de construção via forcing de um espaço compacto e conexo $K$ hereditariamente fracamente Koszmider.This work has two main objectives. First, we analyze the preservation of connectedness in the extension of compact spaces by continuous functions, a technique used by Koszmider to obtain an indecomposable Banach space $C(K)$ with few operators. We show that for any metrizable compactum $K$ there exists a disconnected $L$ which is obtained from $K$ by finitely many extensions by continuous functions. Next, we emphasize the construction of Banach spaces of the form $C(K)$ with the property that $C(L)$ has few operators, for every closed $L \\subseteq K$. Assuming the diamond principle we construct a family $(K_\\xi)_{\\xi < 2^{(2^\\omega)}}$ of connected and hereditarily Koszmider spaces such that every operator from $C(K_\\xi)$ into $C(K_\\eta)$ is weakly compact, for $\\xi$ different from $\\eta$. In particular, $(C(K_\\xi))_{\\xi < 2^{(2^\\omega)}}$ is a family of indecomposable and pairwise essentially incomparable Banach spaces, and each space $K_\\xi$ responds positively to the Efimov\'s problem. We also present a method of construction using forcing of a compact and connected hereditarily weakly Koszmider space $K$.
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Malý, Lukáš. "Sobolev-Type Spaces : Properties of Newtonian Functions Based on Quasi-Banach Function Lattices in Metric Spaces." Doctoral thesis, Linköpings universitet, Matematik och tillämpad matematik, 2014. http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-105616.
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This thesis consists of four papers and focuses on function spaces related to first-order analysis in abstract metric measure spaces. The classical (i.e., Sobolev) theory in Euclidean spaces makes use of summability of distributional gradients, whose definition depends on the linear structure of Rn. In metric spaces, we can replace the distributional gradients by (weak) upper gradients that control the functions’ behavior along (almost) all rectifiable curves, which gives rise to the so-called Newtonian spaces. The summability condition, considered in the thesis, is expressed using a general Banach function lattice quasi-norm and so an extensive framework is built. Sobolev-type spaces (mainly based on the Lp norm) on metric spaces, and Newtonian spaces in particular, have been under intensive study since the mid-1990s. In Paper I, the elementary theory of Newtonian spaces based on quasi-Banach function lattices is built up. Standard tools such as moduli of curve families and the Sobolev capacity are developed and applied to study the basic properties of Newtonian functions. Summability of a (weak) upper gradient of a function is shown to guarantee the function’s absolute continuity on almost all curves. Moreover, Newtonian spaces are proven complete in this general setting. Paper II investigates the set of all weak upper gradients of a Newtonian function. In particular, existence of minimal weak upper gradients is established. Validity of Lebesgue’s differentiation theorem for the underlying metric measure space ensures that a family of representation formulae for minimal weak upper gradients can be found. Furthermore, the connection between pointwise and norm convergence of a sequence of Newtonian functions is studied. Smooth functions are frequently used as an approximation of Sobolev functions in analysis of partial differential equations. In fact, Lipschitz continuity, which is (unlike
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Ghawadrah, Ghadeer. "Théorie descriptive des ensembles et espaces de Banach." Thesis, Paris 6, 2015. http://www.theses.fr/2015PA066078/document.
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Cette thèse traite de la théorie descriptive des ensembles et de la géométrie des espaces de Banach. La première partie consiste en l’étude de la complexité descriptive de la famille des espaces de Banach avec la propriété d’approximation bornée, respectivement la propriété π, dans l’ensemble des sous-espaces fermés de C(Δ), où Δ est l’ensemble de Cantor. Ces familles sont boréliennes. En outre, nous montrons que si alphaThis thesis deals with the descriptive set theory and the geometry of Banach spaces.The first chapter consists of the study of the descriptive complexity of the set of Banachspaces with the Bounded Approximation Property, respectively π-property, in the set ofall closed subspaces of C(∆), where ∆ is the Cantor set. We show that these sets areBorel. In addition, we show that if α<ω_1, the set of spaces with Szlenk index at most α which have a shrinking FDD is Borel. We show in the second chapter that the numberof isomorphism classes of complemented subspaces of the reflexive Orlicz function space L^Φ [0,1] is uncountable, where L^Φ [0,1]is not isomorphic to L^2 [0,1]
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Procházka, Antonín. "Analyse dans les espaces de Banach." Thesis, Bordeaux 1, 2009. http://www.theses.fr/2009BOR13801/document.
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Cette thèse traite quatre sujets différents de la théorie des espaces de Banach: Le premier est une caractérisation de la propriété de Radon-Nikodym en utilisant la notion du jeu des points et tranches: Le deuxième est une évaluation de l'indice de dentabilité préfaible des espaces C(K) où K est un compact du hauteur dénombrable: Le troisième est un renormage des espaces non séparables qui est simultanément LUC, lisse et approximable par des normes d'une lissité plus élevée. Le quatrième est une approche par le théorème de Baire aux principes variationnels paramétriques. La thèse commence par une introduction qui examine le contexte de ces résultatsThe thesis deals with four topics in the theory of Banach spaces. The first of them is a characterization of the Radon-Nikodym property using the notion of point-slice games. The second is a computation of the w* dentability index of the spaces C(K), where K is a compact of countable height. The third is a renorming result in nonseparable spaces, producing norms which are differentiable, LUR and approximated by norms of higher smoothness. The fourth topic is a Baire cathegory approach to parametric smooth variational principles. The thesis features an introduction which surveys the background of these results
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Moreau, Pierre. "Notions de petitesse, géométrie des espaces de Banach et hypercyclicité." Thesis, Bordeaux 1, 2009. http://www.theses.fr/2009BOR13803/document.
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Il existe de nombreuses notions de petitesse en analyse. On considère trois d'entre elles: la Haar-négligeabilité, la Gauss-négligeabilité et la sigma-porosité. On étudie à quelles conditions le cône positif d'une base de Schauder est Haar-négligeable, et ce que cela entraîne pour l'espace de Banach associé. On étudie également sous quelles conditions l'ensemble des vecteurs non-hypercycliques d'un opérateur hypercyclique est Haar-négligeable ou sigma-poreuxThere are many notions of smallness in Analysis. We will consider three of them: Haar-negligeability, Gauss-negligeability and sigma-porosity. We will study on which conditions the positive cone of a Schauder basis is Haar-null, and its consequence on the Banach space. We will also study on which conditions the set of non-hypercyclic vectors of an hypercyclic operator is Haar-null or sigma-porous
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Miyamura, Mauricio Yudi. "Reflexidade de espaços de operadores lineares e espaços de polinomios homogeneos." [s.n.], 2007. http://repositorio.unicamp.br/jspui/handle/REPOSIP/307330.
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Orientador: Jorge Tulio Mujica AscuiDissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica
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Resumo: Sejam E e F espaços de Banach. Os principais resultados que iremos expor serão teoremas sobre a reflexividade de L (E; F) e P (mE; F).. No capítulo 2, estudamos alguns conceitos básicos da teoria de produtos tensoriais de espaços de Banach. A importância do capítulo 2 para o trabalho seria, essencialmente, a identificação do espaço de operadores lineares contínuos L (E; F) com o dual do produto tensorial projetivo E ÄpF?. No capítulo 3, que trata de espaços de polinômios homogêneos, incluímos de noções e resultados básicos e estudamos um teorema de linearização que permitirá transferir resultados em espaços de operadores lineares para espaços de polinômios homogêneos.
Abstract: Let E and F be Banach spaces. The main results in this work are theorems concerning the reflexivity of L (E; F) and P (mE; F). In Chapter 2, we study basic concepts of the theory of tensor products of Banach spaces. The importance of Chapter 2 will be, essentially, the identification of the space of continuous linear operators L(E; F) with the dual of the projective tensor product E ÄpF?. In Chapter 3, that deals with homogeneous polynomials, we include basic definitions and results and we study a linearization theorem that will allow to transfer results from spaces of linear operators to spaces of homogeneous polynomials.
Mestrado
Matematica
Mestre em Matemática
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Augé, Jean-Matthieu. "Quelques problèmes de dynamique linéaire dans les espaces de Banach." Phd thesis, Université Sciences et Technologies - Bordeaux I, 2012. http://tel.archives-ouvertes.fr/tel-00744968.
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Cette thèse est principalement consacrée à des problèmes de dynamique linéaire dans les espaces de Banach. Répondant à une question récente de Hájek et Smith, on construit notamment, dans tout espace de Banach séparable de dimension infinie, un opérateur borné tel que ses orbites tendent vers l'infini sur une partie ni vide, ni dense. On relie également, à l'aide d'un autre résultat, le module de lissité asymptotique au comportement des opérateurs bornés.
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Tocha, Neusa Nogas. "Zeros de polinômios e propriedades polinomiais em espaços de Banach." Universidade de São Paulo, 2006. http://www.teses.usp.br/teses/disponiveis/45/45131/tde-14012007-194635/.
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Neste trabalho temos por objetivo apresentar alguns resultados relacionados aos temas abordados por Aron, Choi e Llavona (1995), Aron e Dimant (2002) e Aron e Rueda (1997). Primeiramente, vamos estudar as propriedades polinomiais (P) e (RP) para os espaços de Banach e a propriedade ACL para as funções definidas entre as bolas unitárias fechadas do espaço. Vamos apresentar novos exemplos de espaços de Banach que possuem a propriedade (P) onde é possível exibir funções que satisfazem a propriedade ACL. Vamos ainda estudar o conjunto de continuidade seqüencial fraca de um polinômio N-homogêneo contínuo com valores vetoriais. Apresentamos as suas propriedades básicas e algumas conexões com o caso dos polinômios escalares. No espaço dual faremos uma breve análise dos polinômios com certo tipo de continuidade com relação à topologia fraca-estrela. Numa outra direção, estudamos os zeros de polinômios N-homogêneos em várias variáveis complexas, mais especificamente, dados n, N números naturais existe um número natural m tal que para cada polinômio N-homogêneo complexo P definido no espaço vetorial C^ existe um subespaço vetorial X_ contido no conjunto dos zeros do polinômio P de dimensão n. Aqui, o principal objetivo é melhorar as limitações para m encontradas por Aron e Rueda (1997) como também generalizar os seus resultados.Our purpose here is to study some results regarding the articles of Aron, Choi and Llavona (1995), Aron and Dimant (2002) and Aron and Rueda (1997). Firstly, we study properties (P) and (RP) for the Banach spaces and the ACL property for the functions defined between the closed unit balls. We give new examples of Banach spaces which have (P) property and some functions defined in those spaces satisfying the ACL property. We also study the set of weak sequential continuity of a vector-valued continuous Nhomogeneous polynomial. In the dual space we study the N-homogeneous polynomials which are weak-star continuous on bounded sets. Finally, we study the zeros of complex N-homogeneous polynomials. This means, given positive integers n and N, there is a positive integer m such that an complex N-homogeneous polynomial P defined in C^ has an ndimensional subspace contained in its zero set. We discuss the problem of finding a good bound on m as a function of n and N. We improve the results given by Aron and Rueda (1997) as also generalize their results.
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Rupprecht, Christoph [Verfasser], Luise [Akademischer Betreuer] Blank, and Michael [Akademischer Betreuer] Hinze. "Projection type methods in Banach space with application in topology optimization / Christoph Rupprecht. Betreuer: Luise Blank ; Michael Hinze." Regensburg : Universitätsbibliothek Regensburg, 2016. http://d-nb.info/1098531167/34.
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Rupprecht, Christoph Verfasser], Luise [Akademischer Betreuer] Blank, and Michael [Akademischer Betreuer] [Hinze. "Projection type methods in Banach space with application in topology optimization / Christoph Rupprecht. Betreuer: Luise Blank ; Michael Hinze." Regensburg : Universitätsbibliothek Regensburg, 2016. http://nbn-resolving.de/urn:nbn:de:bvb:355-epub-337156.
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Yao, Xudong. "Minimax methods for finding multiple saddle critical points in Banach spaces and their applications." Diss., Texas A&M University, 2004. http://hdl.handle.net/1969.1/2732.
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This dissertation was to study computational theory and methods for ?nding multiple saddle critical points in Banach spaces. Two local minimax methods were developed for this purpose. One was for unconstrained cases and the other was for constrained cases. First, two local minmax characterization of saddle critical points in Banach spaces were established. Based on these two local minmax characterizations, two local minimax algorithms were designed. Their ?ow charts were presented. Then convergence analysis of the algorithms were carried out. Under certain assumptions, a subsequence convergence and a point-to-set convergence were obtained. Furthermore, a relation between the convergence rates of the functional value sequence and corresponding gradient sequence was derived. Techniques to implement the algorithms were discussed. In numerical experiments, those techniques have been successfully implemented to solve for multiple solutions of several quasilinear elliptic boundary value problems and multiple eigenpairs of the well known nonlinear p-Laplacian operator. Numerical solutions were presented by their pro?les for visualization. Several interesting phenomena of the solutions of quasilinear elliptic boundary value problems and the eigenpairs of the p-Laplacian operator have been observed and are open for further investigation. As a generalization of the above results, nonsmooth critical points were considered for locally Lipschitz continuous functionals. A local minmax characterization of nonsmooth saddle critical points was also established. To establish its version in Banach spaces, a new notion, pseudo-generalized-gradient has to be introduced. Based on the characterization, a local minimax algorithm for ?nding multiple nonsmooth saddle critical points was proposed for further study.
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Nascimento, Carlos Alberto do. "Estudo sobre espaços de Banach e de Hilbert com aplicações em equações diferenciais, integrais e teoria da aproximação." Universidade Estadual Paulista (UNESP), 2018. http://hdl.handle.net/11449/154152.
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Neste trabalho, abordaremos os principais conceitos e propriedades sobre espaço de Banach e espaço de Hilbert com o objetivo de oferecer o conteúdo necessário para discutirmos algumas aplicações desses conceitos. Mostraremos a existência e unicidade de solução de Equações Diferenciais Ordinárias de Primeira Ordem, existência e unicidade de solução de certas Equações Integrais e existência e unicidade de melhor aproximação em espaços normados e de Hilbert.
In this work, we will discuss the main concepts and properties on Banach space and Hilbert space in order to offer the necessary content to discuss some applications of these concepts. We will show the existence and uniqueness of the solution of First Order Ordinary Differential Equations, existence and uniqueness of solution of certain Integral Equations and existence and uniqueness of better approximation in normed and Hilbert spaces.
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Bertoin, Jean. "Processus de Dirichlet." Grenoble 2 : ANRT, 1987. http://catalogue.bnf.fr/ark:/12148/cb37602956z.
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Assani, Idris. "Contribution à la théorie ergodique des opérateurs dans les espaces ℓp, 1 plus petit ou égal que p plus petit ou égal que plus l'infini : applications multivoques a valeurs dans un espace de Banach." Paris 6, 1986. http://www.theses.fr/1986PA066010.
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Convergence ponctuelle de quelques suites d'opérateurs. Operateurs à puissances bornées et le théorème ergodique ponctuel dans ℓp(0,1), 1