Topology (H) Lecture 14 Lecturer: Zuoqin Wang Time: May 9, 2020
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Normal Convergence
Normal convergence Uniform convergence: A sequence fn converges uniformly on K ⊆ C means 8" > 0 9N 8n ≥ N sup jf(z) − g(z)j < ". z2K 1 1 Weierstrass M-test: If 8z 2 K jfn(z)j ≤ Mn and Mn converges, then fn converges uniformly on K. n=1 n=1 X X Normal convergence: Given a domain Ω ⊆ C, a sequence of functions fn : Ω ! C converges locally uniformly means 8z 2 Ω 9δ > 0 such that the functions fn restricted to Bδ(z) converge uniformly. Heine-Borel theorem implies that locally uniform convergence is equivalent to convergence that is uniform on compact subsets. Topologists call this compact convergence, while complex analysts call it normal convergence. Compact-open topology: Let X and Y be topological spaces and C(X; Y ) be the set of all continuous functions X ! Y . For each compact K ⊆ X and open U ⊆ Y let S = ff: f(K) ⊆ Ug. The topology on C(X; Y ) generated by all such S is called the compact-open topology. In this topology functions are near when their values are close on compact sets. The compact-open topology on C(Ω; C) is exactly the topology of normal convergence (see Theorems XII.7.2 [3], 5.1 [4]). The space of holomorphic functions: Let H(Ω) denote the space of holomorphic functions on Ω. We can construct a metric for the compact-open topology on H(Ω) ⊆ C(Ω; C) by writing Ω as a union of a tower of compact subsets and using a bounded uniform metric on these subsets. -
Exercise Set 13 Top04-E013 ; November 26, 2004 ; 9:45 A.M
Prof. D. P.Patil, Department of Mathematics, Indian Institute of Science, Bangalore August-December 2004 MA-231 Topology 13. Function spaces1) —————————— — Uniform Convergence, Stone-Weierstrass theorem, Arzela-Ascoli theorem ———————————————————————————————– November 22, 2004 Karl Theodor Wilhelm Weierstrass† Marshall Harvey Stone†† (1815-1897) (1903-1989) First recall the following definitions and results : Our overall aim in this section is the study of the compactness and completeness properties of a subset F of the set Y X of all maps from a space X into a space Y . To do this a usable topology must be introduced on F (presumably related to the structures of X and Y ) and when this is has been done F is called a f unction space. N13.1 . (The Topology of Pointwise Convergence 2) Let X be any set, Y be any topological space and let fn : X → Y , n ∈ N, be a sequence of maps. We say that the sequence (fn)n∈N is pointwise convergent,ifforeverypoint x ∈X, the sequence fn(x) , n∈N,inY is convergent. a). The (Tychonoff) product topology on Y X is determined solely by the topology of Y (even if X is a topological X X space) the structure on X plays no part. A sequence fn : X →Y , n∈N,inY converges to a function f in Y if and only if for every point x ∈X, the sequence fn(x) , n∈N,inY is convergent. This provides the reason for the name the topology of pointwise convergence;this topology is also simply called the pointwise topology andisdenoted by Tptc . -
Compact-Open Topology
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 222, 1976 THE MACKEYPROBLEM FOR THE COMPACT-OPENTOPOLOGY BY ROBERT F. WHEELER ABSTRACT. Let CC(T) denote the space of continuous real-valued func- tions on a completely regular Hausdorff space T, endowed with the compact- open topology. Well-known results of Nachbin, Shirota, and Warner characterize those T for which Cc(7') is bornological, barrelled, and infrabarrelled. In this paper the question of when CC(T) is a Mackey space is examined. A convex strong Mackey property (CSMP), intermediate between infrabarrelled and Mackey, is introduced, and for several classes of spaces (including first countable and scattered spaces), a necessary and sufficient condition on T for CC(T) to have CSMP is obtained. Let T be a completely regular Hausdorff space, and let CC(T) denote the space of real-valued continuous functions on T, endowed with the compact-open topology. The problem of relating topological properties of T to linear topo- logical properties of CC(T) has been investigated by many prominent mathe- maticians. For example, Nachbin [22] and Shirota [28] used this approach to find the first example of a barrelled locally convex space which is not borno- logical. Many additional results were furnished by Warner [31] ; in particular, he characterized those spaces T for which CC(T) is complete, separable, or in- frabarrelled. More recently, Buchwalter (many papers, enumerated in [3]), De Wilde and Schmets [6], Haydon [15], [16], and many others have provided new insights into the relationship between T and CC(T). One problem left largely untouched by these authors is the following: find necessary and sufficient conditions on T for CC(T) to be a Mackey space. -
Pointwise Convergence Ofofof Complex Fourier Series
Pointwise convergence ofofof complex Fourier series Let f(x) be a periodic function with period 2 l defined on the interval [- l,l]. The complex Fourier coefficients of f( x) are l -i n π s/l cn = 1 ∫ f(s) e d s 2 l -l This leads to a Fourier series representation for f(x) ∞ i n π x/l f(x) = ∑ cn e n = -∞ We have two important questions to pose here. 1. For a given x, does the infinite series converge? 2. If it converges, does it necessarily converge to f(x)? We can begin to address both of these issues by introducing the partial Fourier series N i n π x/l fN(x) = ∑ cn e n = -N In terms of this function, our two questions become 1. For a given x, does lim fN(x) exist? N→∞ 2. If it does, is lim fN(x) = f(x)? N→∞ The Dirichlet kernel To begin to address the questions we posed about fN(x) we will start by rewriting fN(x). Initially, fN(x) is defined by N i n π x/l fN(x) = ∑ cn e n = -N If we substitute the expression for the Fourier coefficients l -i n π s/l cn = 1 ∫ f(s) e d s 2 l -l into the expression for fN(x) we obtain N l 1 -i n π s/l i n π x/l fN(x) = ∑ ∫ f(s) e d s e n = -N(2 l -l ) l N = ∫ 1 ∑ (e-i n π s/l ei n π x/l) f(s) d s -l (2 l n = -N ) 1 l N = ∫ 1 ∑ ei n π (x-s)/l f(s) d s -l (2 l n = -N ) The expression in parentheses leads us to make the following definition. -
Uniform Convexity and Variational Convergence
Trudy Moskov. Matem. Obw. Trans. Moscow Math. Soc. Tom 75 (2014), vyp. 2 2014, Pages 205–231 S 0077-1554(2014)00232-6 Article electronically published on November 5, 2014 UNIFORM CONVEXITY AND VARIATIONAL CONVERGENCE V. V. ZHIKOV AND S. E. PASTUKHOVA Dedicated to the Centennial Anniversary of B. M. Levitan Abstract. Let Ω be a domain in Rd. We establish the uniform convexity of the Γ-limit of a sequence of Carath´eodory integrands f(x, ξ): Ω×Rd → R subjected to a two-sided power-law estimate of coercivity and growth with respect to ξ with expo- nents α and β,1<α≤ β<∞, and having a common modulus of convexity with respect to ξ. In particular, the Γ-limit of a sequence of power-law integrands of the form |ξ|p(x), where the variable exponent p:Ω→ [α, β] is a measurable function, is uniformly convex. We prove that one can assign a uniformly convex Orlicz space to the Γ-limit of a sequence of power-law integrands. A natural Γ-closed extension of the class of power-law integrands is found. Applications to the homogenization theory for functionals of the calculus of vari- ations and for monotone operators are given. 1. Introduction 1.1. The notion of uniformly convex Banach space was introduced by Clarkson in his famous paper [1]. Recall that a norm ·and the Banach space V equipped with this norm are said to be uniformly convex if for each ε ∈ (0, 2) there exists a δ = δ(ε)such that ξ + η ξ − η≤ε or ≤ 1 − δ 2 for all ξ,η ∈ V with ξ≤1andη≤1. -
Compactness Result and Its Applications in Integral Equations
Compactness result and its applications in integral equations Mateusz Krukowski1 and Bogdan Przeradzki2 1,2 Technical University of Łódź, Institute of Mathematics May 12, 2015 Abstract A version of Arzelà-Ascoli theorem for X being σ-locally compact Hausdorff space is proved. The result is used in proving compactness of Fredholm, Ham- merstein and Urysohn operators. Two fixed point theorems, for Hammerstein and Urysohn operator, are derived on the basis of Schauder theorem. Keywords : Arzelà-Ascoli theorem, compact operators, Fredholm, Hammerstein, Urysohn 1 Introduction Compactness criteria in typical function spaces not only constitute important results describing properties of these spaces, but they also give a basic tool for in- vestigating the existence of solutions to nonlinear equations of many kinds. The best known criterion is the Arzelà-Ascoli theorem that gives necessary and sufficient con- ditions for compactness in the space of continuous functions defined on a compact space X and taking values in R or, more generally, in any finite-dimensional Banach space E : a subset K ⊂ C(X, E) should consist of equibounded and equicontinuous functions. It is easy to drop the assumption on the dimension of E but then the functions from K should be pointwise relatively compact, i.e. Kx := {f(x) : f ∈ K} is relatively compact for any x ∈ X. The natural topology in C(X, E) is the topology of uniform convergence given by the norm kfk := supx∈X kf(x)kE. arXiv:1505.02533v1 [math.FA] 11 May 2015 If X is not a compact space but only a locally compact one, the Arzelaà-Ascoli theorem gives a compactness criterion in the space of continuous functions C(X,Y ), where Y is a metric space, with the topology of compact convergence, see [5] (or a compact-open topology [3]). -
Uniform Convergence
2018 Spring MATH2060A Mathematical Analysis II 1 Notes 3. UNIFORM CONVERGENCE Uniform convergence is the main theme of this chapter. In Section 1 pointwise and uniform convergence of sequences of functions are discussed and examples are given. In Section 2 the three theorems on exchange of pointwise limits, inte- gration and differentiation which are corner stones for all later development are proven. They are reformulated in the context of infinite series of functions in Section 3. The last two important sections demonstrate the power of uniform convergence. In Sections 4 and 5 we introduce the exponential function, sine and cosine functions based on differential equations. Although various definitions of these elementary functions were given in more elementary courses, here the def- initions are the most rigorous one and all old ones should be abandoned. Once these functions are defined, other elementary functions such as the logarithmic function, power functions, and other trigonometric functions can be defined ac- cordingly. A notable point is at the end of the section, a rigorous definition of the number π is given and showed to be consistent with its geometric meaning. 3.1 Uniform Convergence of Functions Let E be a (non-empty) subset of R and consider a sequence of real-valued func- tions ffng; n ≥ 1 and f defined on E. We call ffng pointwisely converges to f on E if for every x 2 E, the sequence ffn(x)g of real numbers converges to the number f(x). The function f is called the pointwise limit of the sequence. According to the limit of sequence, pointwise convergence means, for each x 2 E, given " > 0, there is some n0(x) such that jfn(x) − f(x)j < " ; 8n ≥ n0(x) : We use the notation n0(x) to emphasis the dependence of n0(x) on " and x. -
Pointwise Convergence for Semigroups in Vector-Valued Lp Spaces
Pointwise Convergence for Semigroups in Vector-valued Lp Spaces Robert J Taggart 31 March 2008 Abstract 2 Suppose that {Tt : t ≥ 0} is a symmetric diusion semigroup on L (X) and denote by {Tet : t ≥ 0} its tensor product extension to the Bochner space Lp(X, B), where B belongs to a certain broad class of UMD spaces. We prove a vector-valued version of the HopfDunfordSchwartz ergodic theorem and show that this extends to a maximal theorem for analytic p continuations of {Tet : t ≥ 0} on L (X, B). As an application, we show that such continuations exhibit pointwise convergence. 1 Introduction The goal of this paper is to show that two classical results about symmetric diusion semigroups, which go back to E. M. Stein's monograph [24], can be extended to the setting of vector-valued Lp spaces. Suppose throughout that (X, µ) is a positive σ-nite measure space. Denition 1.1. Suppose that {Tt : t ≥ 0} is a semigroup of operators on L2(X). We say that (a) the semigroup {Tt : t ≥ 0} satises the contraction property if 2 q (1) kTtfkq ≤ kfkq ∀f ∈ L (X) ∩ L (X) whenever t ≥ 0 and q ∈ [1, ∞]; and (b) the semigroup {Tt : t ≥ 0} is a symmetric diusion semigroup if it satises 2 the contraction property and if Tt is selfadjoint on L (X) whenever t ≥ 0. It is well known that if 1 ≤ p < ∞ and 1 ≤ q ≤ ∞ then Lq(X) ∩ Lp(X) is p 2 dense in L (X). Hence, if a semigroup {Tt : t ≥ 0} acting on L (X) has the p contraction property then each Tt extends uniquely to a contraction of L (X) whenever p ∈ [1, ∞). -
Topology (H) Lecture 4 Lecturer: Zuoqin Wang Time: March 18, 2021
Topology (H) Lecture 4 Lecturer: Zuoqin Wang Time: March 18, 2021 CONVERGENCE AND CONTINUITY 1. Convergence in topological spaces { Convergence. As we have mentioned, topological structure is created to extend the conceptions of convergence and continuous map to more general setting. It is easy to define the conception of convergence of a sequence in any topological spaces.Intuitively, xn ! x0 means \for any neighborhood N of x0, eventually the sequence xn's will enter and stay in N". Translating this into the language of open sets, we can write Definition 1.1 (convergence). Let (X; T ) be a topological space. Suppose xn 2 X and x0 2 X: We say xn converges to x0, denoted by xn ! x0, if for any neighborhood A of x0, there exists N > 0 such that xn 2 A for all n > N. Remark 1.2. According to the definition of neighborhood, it is easy to see that xn ! x0 in (X; T ) if and only if for any open set U containing x0, there exists N > 0 such that xn 2 U for all n > N. To get a better understanding, let's examine the convergence in simple spaces: Example 1.3. (Convergence in the metric topology) The convergence in metric topology is the same as the metric convergence: xn !x0 ()8">0, 9N >0 s.t. d(xn; x0)<" for all n>N. Example 1.4. (Convergence in the discrete topology) Since every open ball B(x; 1) = fxg, it is easy to see xn ! x0 if and only if there exists N such that xn = x0 for all n > N. -
Almost Uniform Convergence Versus Pointwise Convergence
ALMOST UNIFORM CONVERGENCE VERSUS POINTWISE CONVERGENCE JOHN W. BRACE1 In many an example of a function space whose topology is the topology of almost uniform convergence it is observed that the same topology is obtained in a natural way by considering pointwise con- vergence of extensions of the functions on a larger domain [l; 2]. This paper displays necessary conditions and sufficient conditions for the above situation to occur. Consider a linear space G(5, F) of functions with domain 5 and range in a real or complex locally convex linear topological space £. Assume that there are sufficient functions in G(5, £) to distinguish between points of 5. Let Sß denote the closure of the image of 5 in the cartesian product space X{g(5): g£G(5, £)}. Theorems 4.1 and 4.2 of reference [2] give the following theorem. Theorem. If g(5) is relatively compact for every g in G(5, £), then pointwise convergence of the extended functions on Sß is equivalent to almost uniform converqence on S. When almost uniform convergence is known to be equivalent to pointwise convergence on a larger domain the situation can usually be converted to one of equivalence of the two modes of convergence on the same domain by means of Theorem 4.1 of [2]. In the new formulation the following theorem is applicable. In preparation for the theorem, let £(5, £) denote all bounded real valued functions on S which are uniformly continuous for the uniformity which G(5, F) generates on 5. G(5, £) will be called a full linear space if for every/ in £(5, £) and every g in GiS, F) the function fg obtained from their pointwise product is a member of G (5, £). -
Epigraphical and Uniform Convergence of Convex Functions
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 348, Number 4, April 1996 EPIGRAPHICAL AND UNIFORM CONVERGENCE OF CONVEX FUNCTIONS JONATHAN M. BORWEIN AND JON D. VANDERWERFF Abstract. We examine when a sequence of lsc convex functions on a Banach space converges uniformly on bounded sets (resp. compact sets) provided it converges Attouch-Wets (resp. Painlev´e-Kuratowski). We also obtain related results for pointwise convergence and uniform convergence on weakly compact sets. Some known results concerning the convergence of sequences of linear functionals are shown to also hold for lsc convex functions. For example, a sequence of lsc convex functions converges uniformly on bounded sets to a continuous affine function provided that the convergence is uniform on weakly compact sets and the space does not contain an isomorphic copy of `1. Introduction In recent years, there have been several papers showing the utility of certain forms of “epi-convergence” in the study of optimization of convex functions. In this area, Attouch-Wets convergence has been shown to be particularly important (we say the sequence of lsc functions fn converges Attouch-Wets to f if d( , epifn)converges { } · uniformly to d( , epif) on bounded subsets of X R, where for example d( , epif) denotes the distance· function to the epigraph of×f). The name Attouch-Wets· is associated with this form of convergence because of the important contributions in its study made by Attouch and Wets. See [1] for one of their several contributions and Beer’s book ([3]) and its references for a more complete list. Several applications of Attouch-Wets convergence in convex optimization can be found in Beer and Lucchetti ([8]); see also [3]. -
Topology (H) Lecture 12 Lecturer: Zuoqin Wang Time: April 19, 2021
Topology (H) Lecture 12 Lecturer: Zuoqin Wang Time: April 19, 2021 THE ARZELA-ASCOLI THEOREM 1. Five topologies on C(X; Y ) { Shortcoming of the three topologies. Let X be a topological space, and (Y; d) a metric space. Last time we have seen three topologies, Tp:c:; Tuniform; Tbox, on the space of continuous maps, C(X; Y ) = ff 2 M(X; Y ) j f is continuousg: We want to study convergence of sequences of functions in C(X; Y ). Example 1.1. Consider the case X = Y = R, then (1) With respect to the pointwise convergence topology, the sequence of functions −nx2 fn(x) = e converges in Tp:c: to a bad limit function, the discontinuous function f0(x) which equals 1 at x = 0 and equals 0 for all other x. Underlying reason: The pointwise convergence topology (=the prod- uct topology) is too weak for the limit of a convergent sequence of continuous functions to be continuous. (2) With respect to the uniform convergence topology and the box topology, the 2 sequence of functions fn(x) = x =n would not converge in Tu:c:, although it does converge to a nice limit function f0(x) ≡ 0 in the pointwise sense. Underlying reason: The uniform topology (and thus the box topology) is too strong for a sequence to converge. We want to find a reasonable topology on C(X; Y ) so that \bad convergent se- quences" are no longer convergent in this topology, while \good convergent sequences" are still convergent. By the analysis above, what we need should be a new topology on C(X; Y ) that is weaker than Tuniform, but the limit of a convergent sequence of continuous functions with respect to this new topology is still continuous.