Sokal uniform boundedness. Abstract: In this paper our aim is to expound the importance of Non-Baire Proof of Uniform Boundedness theorem. FA] 29 Dec 2010 A really simple elementary proof of the uniform boundedness theorem Alan D. It provides free access to secondary information on researchers, articles, patents, etc. monthly. Sean X un espacio de Banach, Y un espacio normado y (Tn)n∈N una sucesi ́on en B(X, Y ). Indeed, the basic idea of the 5 A good example for this is Sokal's A Really Simple Elementary Proof of the Uniform Boundedness Theorem, The American Mathematical Monthly Vol. Monthly 118 (2011), No. 6 replaces the linearity assumption with some superadditivity and some scaling assumption. Our proof of Theorem 1 is not only new but also very short. Abstract In this paper our aim is to expound the importance of Non-Baire Proof of Uniform Boundedness theorem. Jun 6, 2020 · The concept of uniform boundedness from below and above has been generalized to the case of mappings $ f: X \rightarrow Y $ into a set $ Y $ that is ordered in some sense. IfFis pointwise bounded, then it is bounded. Dec 27, 2019 · Lecture 20: Uniform boundedness principle Claudio Landim Previous lectures: http://bit. 1 (Faltings). 5 (May 2011), pp. Jan 1, 2015 · Citations (0) References (1) A Really Simple Elementary Proof of the Uniform Boundedness Theorem Article May 2010 Alan Sokal Nonlinear Systems and Control Lecture # 18 Boundedness & Ultimate Boundedness Definition: The solutions of ̇x = f (t, x) are uniformly bounded if ∃ ∃ lter F on N; we introduce and study corresponding uniform F-boundedness principles for locally convex topological vector spaces. Sokal gave a very short and completely elementary proof of the uniform boundedness principle. $ (T_ {i})_ {i\in I}$ be a family of bounded linear operators between $\mathcal {X}$ and $\mathcal {Y}$. In this paper we show that these three pillars should be either just two or at least eight, since the Nov 21, 2023 · A Really Simple Elementary Proof of the Uniform Boundedness Theorem Article May 2010 Alan Sokal 3 Corolario (teorema de Banach–Steinhaus sobre las sucesiones puntualmente conver-gentes de operadores lineales acotados). Sokal gave a very short and completely elementary proof of the uniform boundedness princi-ple. If is pointwise bounded (i. " Symbolically, if is finite for each in the unit ball, then is finite. Steinhaus known as the Banach-Steinhaus theorem or Uniform boundedness principle: a pointwise-bounded family of continuous linear operators from a Banach space to a normed space is uniformly bounded. My book say the same thing but iff $\\mathcal F$ is local On uniform boundedness of sequential social learning Itay Kavalera,∗ aDavidson Faculty of Industrial Engineering and Management, Technion, Haifa 3200003, Israel. With that claim it's easy to prove Banach-Steinhaus. Rudin's proof is beautiful and I think, sorry if I'm wrong, it's the most general one. Mar 1, 2019 · Compared to the statements of the classical uniform boundedness principle in Banach spaces, the nonlinear uniform boundedness principle of Theorem 1. Sokal 0 references publication date 2011 0 references published in American Mathematical Monthly 0 references In [12], Alan D. , in science and technology, medicine and pharmacy. J. We know that Baire’s Category Theorem is directly applied to prove Uniform Boundedness Theorem. The American Mathematical Monthly, 118 (5), 450. Uniform boundedness principle was discovered by Lebesgue in 1908 in investigations on Fourier series, it was isolated as a general principle by Banach and Steinhaus. Nov 27, 2020 · In this article we introduce a dual of the uniform boundedness principle which does not require completeness and gives an indirect means for testing the boundedness of a set. does not use any version of the Baire category theorem) and also extremely simple. The aim of this note is to give a simple new proof of Theorem 1 using the well-known uniform boundedness principle, which we state as Theorem 2, and a new result, stated as Theorem 3, which is proved in Section 2. Abstract I give a proof of the uniform boundedness theorem that is elementary (i. Jan 18, 2025 · I hope to understand a (yet another) proof of the uniform boundedness principle, which states that: let $\mathcal {X}$ be a real Banach space, $\mathcal {Y}$ be a real normed vector space. The signifiance of this theorem is that it shows that pointwise control implies uniform control. The aim of this paper is Oct 20, 2010 · I give a proof of the uniform boundedness theorem that is elementary (i. 118. Perhaps the simplest example is the theorem that a continuous function on a compact set is uniformly contin-uous. In its basic form, it asserts that for a family of continuous linear operators (and thus bounded operators) whose domain is a Banach Article "A Really Simple Elementary Proof of the Uniform Boundedness Theorem" Detailed information of the J-GLOBAL is an information service managed by the Japan Science and Technology Agency (hereinafter referred to as "JST"). This document provides a simple elementary proof of the uniform boundedness theorem. Sokal. a Banach space a normed linear space a family of bounded linear operators from to . Alan D. Sokal gives an elementary proof of the uniform boundedness theorem, utilizing the following lemma: Lemma. It shows that if a family of operators is bounded at each point, it's actually bounded everywhere. Our approach is an adaptation to Isabelle/HOL of a proof due to A. F. Uniform boundedness theorem, open mapping theorem, closed graph theorem. May 10, 2010 · Recently Alan D. Stated another way, let be a Banach space and be a normed space. 05. Banach and H. Steinhaus [1] known as Banach-Steinhaus theorem or Uniform bound-edness principle: a pointwise-bounded family of continuous linear op-erators from a Banach space to a normed space is uniformly bounded. The Banach-Steinhaus theorem tells us that \second category" is a su cient condition, but the Nikodym boundedness theorem shows that a uniform boundedness principle is true under weaker conditions, in particular spaces at least. The search results guide you Alan SOKAL | Cited by 13,881 | of University College London, London (UCL) | Read 235 publications | Contact Alan SOKAL Abstract I give a proof of the uniform boundedness theorem that is elementary (i. 118, No. The first one is derived from Zorn’s Lemma, while the latter two usually are obtained from Baire’s Category Theorem. All these are consequences of a topological result known as Baire's (category) theorem. I found this proof in Emmanuele DiBenedetto: Real Analysis. , does not use any version of the Baire category theorem) and also extremely simple. Nov 19, 2014 · Wikipedia gives Montel's theorem saying essentially $\\mathcal F$ is a normal family in $\\mathbb H(G)$ iff $\\mathcal F$ is uniformly bounded. It then presents a new elementary proof that constructs a sequence to find a point where the supremum is infinite, avoiding the need for Baire category. Aug 1, 2025 · In this paper, Alan D. 1585v2 [math. In its basic form, it asserts that for a family of continuous linear operators (and thus bounded operators) whose domain is a Banach Mar 13, 2020 · A Really Simple Elementary Proof of the Uniform Boundedness Theorem Article May 2010 Alan Sokal We will now reproduce the proof (due to Sokal [5]), presented in Kesavan [2], of the uniform boundedness theorem, which does not use the Baire category theorem. 5, 450-452, gave a very short and completely elementary proof of the uniform boundedness principle. , for all ), then is norm-bounded (i. e. Sokal Uniform boundedness theorem. ly/2Z3qzIM These lectures are mainly based on the book "Functional Analysis" by Peter Lax. 450–452, ArXiV Version. Dec 5, 2020 · Uniform boundedness principle was discovered by Lebesgue in 1908 in investigations on Fourier series, it was isolated as a general principle by Banach and Steinhaus. This proof proceeds by induction and is of is extremely wasteful from a quantitative point of view. Math. , 19, 155–190 (1897). Theorem 1. While admittedly it is beautifully simple and elementary, it involves a plain application of dependent choice in the main argument. In its basic form, it asserts that for a family of continuous linear operators (and thus bounded operators) whose domain is a Banach Remark. Let X be a smooth, projective curve defined over a number field K of genus The uniform boundedness principle, also known as the Banach-Steinhaus theorem, states that for a pointwise convergent sequence of continuous linear functionals on a Fréchet space, there exists a uniform bound on the norms of the functionals. Feb 6, 2024 · In mathematics, the uniform boundedness principle or Banach–Steinhaus theorem is one of the fundamental results in functional analysis. Theory B 103 (1): 21-45 (2013) 2011 [j10] Alan D. Makay published Uniform boundedness and uniform ultimate boundedness for functional differential equations | Find, read and cite all the research you need on ResearchGate This variant of the Banach-Steinhaus (uniform boundedness) theorem is used with Banach-Alaoglu to show that weak boundedness implies boundedness in a locally convex space, the starting point for weak-to-strong principles. 9 (uniform boundedness theorem)LetXbe a Banach space andYa normed linear space, and letFbe a family of bounded linear maps fromXtoY. 118 (5): 450-452 (2011) 2010 [j9] Abstract Three grand theorems of functional analysis are the uniform boundedness (or, Banach-Steinhaus) theorem, the open mapping theorem and the closed graph theorem. The dual principle 6 days ago · Uniform Boundedness Principle A "pointwise-bounded" family of continuous linear operators from a Banach space to a normed space is "uniformly bounded. ) See also Sokal’s A really simple elementary proof of the Mar 1, 2021 · For every filter F on N, we introduce and study corresponding uniform F -boundedness principles for locally convex topological vector spaces. Ansorena Abstract. Comb. We give a proof of the uniform boundedness principle for linear continuous maps from F-spaces into topological vector spaces which is elemen-tary and also quite simple. All three rely on the completeness of some or all of the spaces involved, and their proofs are based on Baire's theorem (or, the Baire category theorem), a topological conclusion. Uniform boundedness for arbitrary families Is it true that in any algebraic family of varieties, the number of rational points of the varieties is uniformly bounded after discarding the varieties with in nitely many rational points? Aug 21, 2021 · Many authors consider that the main pillars of Functional Analysis are the Hahn–Banach Theorem, the Uniform Boundedness Principle and the Open Mapping Principle. To see this, let F be a family of The uniform boundedness theorem is one of the central theorems of functional analysis and it has first been published in Banach’s thesis, in the year 1922. Together with the Hahn–Banach theorem and the open mapping theorem, it is considered one of the cornerstones of the field. It's used to prove other important theorems, like the Banach-Steinhaus Theorem, and helps analyze The Principle of Uniform Boundedness, and Friends In these notes, unless otherwise stated, X and Y are Banach spaces and T : X → Y is linear and has domain X . Jul 13, 2025 · The proving of three big theorems, known as the uniform boundedness theorem, the open mapping theorem, and the closed graph theorem, is the pinnacle of any first functional analysis 58 The uniform boundedness principle of Functional Analysis is a very important application of the Baire Category Theorem. Dec 13, 2017 · I give a proof of the uniform boundedness theorem that is elementary (i. This principle has far-reaching consequences. I give a proof of the uniform boundedness theorem that is elementary (i. In its basic form, it asserts that for a family of continuous linear operators (and thus bounded operators) whose domain is a Banach Jan-David Hardtke Abstract. Entonces, S ∈ B(X, Y ). The proving of three big theorems, known as the uniform boundedness theorem, the open mapping theorem, and the closed graph theorem, is the pinnacle of any first functional analysis course. Faltings’ Theorem. This powerful result connects pointwise and uniform behavior of operators. A quantita-tively sharp version of the uniform boundedness theorem follows from Ball’s “plank theorem” [1]: namely, if P∞ kTnk−1 < ∞, then there exists n 1 th fu May 10, 2010 · I give a proof of the uniform boundedness theorem that is elementary (i. This principle connects deeply with concepts like continuity and convergence, influencing results in functional analysis and providing insights into operator behavior in The Principle of Uniform Boundedness, and Friends In these notes, unless otherwise stated, X and Y are Banach spaces and T : X → Y is linear and has domain X . 450 I give a proof of the uniform boundedness theorem that is elementary (i. 1. A Really Simple Elementary Proof of the Uniform Boundedness Theorem. Sokal [3]. The most basic examples of normed spaces are, of course, the scalar fields R and C. Uniform boundedness In mathematics, a uniformly bounded family of functions is a family of bounded functions that can all be bounded by the same constant. Mon. We formalize in Isabelle/HOL a result due to S. doi:10. It is perhaps interesting to note that the uniform boundedness principle for linear functionals (i. Apr 16, 2024 · 数学 上， 一致有界性原理，又称 巴拿赫–斯坦豪斯定理[1] 、 共鸣定理，是 泛函分析 的重要结果。定理断言，对于任意一族定义在 巴拿赫空间 上的 连续线性算子，该族算子逐点有界，当且仅当其在 算子范数 意义下一致有界。 定理最早由 斯特凡·巴拿赫 和 胡戈·斯坦豪斯 于1927年发表，亦由 I do not know Brezis' proof, but on complete spaces an immediately consequence of Baire Category Thm is the uniform boundedness (on an open ball) of continuous bounded real valued functions from your complete Space. implies uniform boundedness. View of Non-Baire Proof of Uniform Boundedness Theorem and Its Applications in the Proof of Some Grand Theorems of Functional Analysis In mathematics, the uniform boundedness principle or Banach–Steinhaus theorem is one of the fundamental results in functional analysis. INTRODUCTION 1. Let fTigi2A either there exists M 0 such that sup Ti∥ i 2A ∥ Functional Analysis 24 | Uniform Boundedness Principle / Banach–Steinhaus Theorem The Bright Side of Mathematics 207K subscribers 398 The Uniform Boundedness Principle Recall from The Lemma to the Uniform Boundedness Principle page that if is a complete metric space and is a collection of continuous functions on then if for each , then there exists a nonempty open set such that: (1) This important result is known as the uniform boundedness theorem (or principle of uniform boundedness or Banach–Steinhaus theorem):8 Theorem 7. In its basic form, it asserts that for a family of continuous linear operators (and thus bounded operators) whose domain is a Banach A Really Simple Elementary Proof of the Uniform Boundedness Theorem(English) 0 references main subject mathematical proof 1 reference based on heuristic inferred from title author Alan Sokal series ordinal 1 object named as Alan D. Alan David Sokal (/ ˈsoʊkəl / SOH-kəl; born January 24, 1955) is an American professor of mathematics at University College London and professor emeritus of physics at New York University. May 10, 2010 · I give a proof of the uniform boundedness theorem that is elementary (i. Uniform Boundedness Principle— Let be a Banach space, a normed vector space and the space of all continuous linear operators from into . Thus all the three grand theorems can be proved without using Baire's theorem. Sokal in Amer. Sokal, A really simple elementary proof of the uniform boundedness theorem Jan 1, 1995 · PDF | On Jan 1, 1995, G. Sokal gave a very short and elegant proof of the classical uniform boundedness principle for linear operators on Banach spaces, which is also completely elementary (in particular, it does not use the Baire category theorem). It begins by stating the uniform boundedness theorem and noting that standard proofs rely on the Baire category theorem. The aim of this note is to point out that by using a similiar technique one can give a considerably short and simple proof of a stronger statement, namely a principle of condensation of sin-gularities for certain double-sequences of non-linear Sokal [1] による，一様有界性原理のBaireのカテゴリー定理を経由しないシンプルな証明を紹介します． Abstract I give a proof of the uniform boundedness theorem that is elementary (i. These principles generalise the classical uniform boundedness principles for se-quences of continuous linear maps by coinciding with these principles when the lter F equals the Frechet lter of co nite subsets of N: We determine combinatorial properties for the lter F Home Uniform Boundedness Theorem Preview Full text arXiv:1005. Apr 12, 2025 · Calculus and Analysis Functional Analysis Principle of Uniform Boundedness See Uniform Boundedness Principle Jun 14, 2025 · Embark on a journey through the realm of functional analysis and discover the intricacies of the Uniform Boundedness Principle in Topological Vector Spaces. The uniform boundedness theorem is one of the central theorems of functional analysis and it has first been published in Banach’s thesis, in the year 1922. We determine combinatorial lter F on N; we introduce and study corresponding uniform F-boundedness principles for locally convex topological vector spaces. These principles generalise the classical uniform boundedness principles for se-quences of continuous linear maps by coinciding with these principles when the lter F equals the Frechet lter of co nite subsets of N: We determine combinatorial properties for the lter F The purpose of this note is to present an alternative proof of the uniform bound-edness theorem, without the need for the Baire category theorem. Sokal Abstract. Given a connected Riemannian manifold N, an m -dimensional Riemannian manifold M which is either compact or the Euclidean space, p ∈ [1, + ∞) and s ∈ (0, 1], we establish, for the problems of surjectivity of the trace, of weak-bounded approximation, of lifting and of superposition, that qualitative properties satisfied by every map in a nonlinear Sobolev space W s, p (M, N) imply In mathematics, the uniform boundedness principle or Banach–Steinhaus theorem is one of the fundamental results in functional analysis. Let $ X $ be a linear topological space that is not a countable union of closed nowhere-dense subsets. The theorem is a corollary of the Banach-Steinhaus theorem. Abstract Many authors consider that the main pillars of Functional Analysis are the Hahn–Banach Theorem, the Uniform Boundedness Principle and the Open Mapping Principle. Suppose that is a collection of continuous linear operators from to If, for every , then Jan 23, 2024 · Arxiv: Alan D. Mathematics Subject Classification (MSC 2000) codes: 46B20, 46B28 (Secondary). quantitative we have the qualitative statement that the family of operators is somehow well-behaved, and the May 11, 2010 · Alan D. Complex zero-free regions at large |q| for multivariate Tutte polynomials (alias Potts-model partition functions) with general complex edge weights. 4169/amer. These principles generalise the classical uniform boundedness principles for sequences of continuous linear maps by coinciding with these principles when the filter F equals the Fréchet filter of cofinite subsets of N. The Uniform Boundedness Principle states that for a collection of continuous linear operators from a Banach space to a normed space, if each operator is pointwise bounded on the space, then the collection is uniformly bounded. Uniform boundedness principle was I give a proof of the uniform boundedness theorem that is elementary (i. math. Comments The uniform boundedness theorem is as follows. b. The main theorem in this section concerns a family of bounded lin-ear operators, and asserts that the family is uniformly bounded (and History of SokolSokol Utilized Various Facilities as they Established their ClubsSound Mind in a Sound BodyDemocracyBrotherhood FellowshipEthnic IdentityEqualityCivic ResponsibilityCzech Autonomy for two centuries (1306- 1525): Pre-Habsburg Rule and its role in eventual Czech freedom Czechs (Bohemians and Moravians) enjoyed virtual self-rule, especially under King Charles IV who became a Holy On Banach-Steinhaus Theorem (Uniform Boundedness Theorem) f the Bana m 1. One of the pillars of functional analysis is the uniform boundedness theorem: Uniform Boundedness Theorem. Our main result in this paper is that \not thin" is the condition sought for this problem as well. José L. (2011). Sep 17, 2015 · Sokal's short proof of the uniform boundedness principle Ask Question Asked 10 years, 1 month ago Modified 1 year, 8 months ago Alan D. This story begins with the following theorem proved by Faltings in the 80s, which is one of the crowning achievements of 20th century arithmetic geometry. Let X be a Banach space a d let Y be a normed space. Problem Sets: One cannot learn mathematics solely by watching someone else do math- ematics (even if that “someone” is a UCL professor). , in the special case that the codomain is a scalar field) implies the same theorem for all linear operators using the above remark about the computation of norms. math Abstract We give a proof of the uniform boundedness principle for linear continuous maps from F -spaces into topological vector spaces which is elementary and also quite simple. Key Words: Uniform boundedness; gliding hump; sliding hump; Baire category. Supongamos que para cada x en X la suce-si ́on (Tnx)n∈N converge en Y a un vector que denotemos por S(x). From the perspective of qualitative vs. In [2] a proof of Theorem 1 is given, which is different from ours. Osgood: Nonuniform convergence and the integration of se-ries term by term, Amer. In mathematics, the uniform boundedness principle or Banach–Steinhaus theorem is one of the fundamental results in functional analysis. We present here, a proof of the Uniform Boundedness Theorem that does not require the Baire’s Theorem in a similar fashion as proved by Alan D Sokal [1] but in a slightly different way. Sokal∗ Department of Physics New York University 4 Washington Place New York, NY 10003 USA [email protected] May 7, 2010 revised October 20, 2010 to appear in the American Mathematical Monthly In mathematics, the uniform boundedness principle or Banach–Steinhaus theorem is one of the fundamental results in functional analysis. The aim of this note is to point out that by using a similiar technique one can give a considerably short and simple proof of a stronger statement, namely a principle of condensation of singularities for certain double-sequences of non-linear operators on quasi-Banach spaces, which is Abstract We formalize in Isabelle/HOL a result [2] due to S. Added: (t. Am. Jun 6, 2024 · We give a proof of the uniform boundedness principle for linear continuous maps from F-spaces into topological vector spaces which Abstract I give a proof of the uniform boundedness theorem that is elementary (i. May 10, 2010 · I give a proof of the uniform boundedness theorem that is elementary (i. Since you said that the uniform boundedness principle is still a bit of a mystery to you, I can't do better than recommend Alan Sokal's recent article A really simple elementary proof of the uniform boundedness theorem in which he gives a proof that gets away without using any Baire-trickery. The Uniform Boundedness Principle is a key concept in functional analysis. May 27, 2025 · Dive into the world of real analysis and explore the Uniform Boundedness Principle, a fundamental concept in functional analysis. The mathematical literature has several results which show the interconnections between these theorems, some well known and well documented Apr 10, 2012 · Recently Alan D. DiBenedetto refers to an article by W. We know that Baire s Category Theorem is directly applied to prove Uniform Boundedness Theorem. , ). This constant is larger than or equal to the absolute value of any value of any of the functions in the family. Since, for our purposes at least, \not Note that the condition that \ (\ { || T_\alpha || : \alpha \in A \}\) is equivalent to the family \ (\ { T_\alpha \}\) to be equicontinuous. Sokal: A Really Simple Elementary Proof of the Uniform Boundedness Theorem. We present here, a proof of the Uniform Boundedness Theorem that does not require the Baire s Theorem in a similar fashion as proved by Alan D Sokal [1] but in a slightly different way. Recently in [11] Alan D. Let $T$ be a bounded linear operator Nov 8, 2023 · A Really Simple Elementary Proof of the Uniform Boundedness Theorem 1 May 2011AMERICAN MATHEMATICAL MONTHLY118 (5):450-452 (3 pages)MATHEMATICAL ASSOC AMER Sokal AD VIEW MORE INFO 10. In this paper we show that these three pillars should be either just two or at least eight, since the Uniform By the foregoing observations, it is plain that, for each in , the sequence UNIFORM BOUNDEDNESS OF RATIONAL POINTS TONY FENG 1. 1 The Principal of Uniform Boundedness Many of the most important theorems in analysis assert that pointwise hy-potheses imply uniform conclusions. ezrnytza2wpkxnc7ncuwjr3fazrvoa0kvzzmypujocnnblnle3ateiq