Introduction to Banach Spaces
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
FUNCTIONAL ANALYSIS 1. Metric and Topological Spaces A
FUNCTIONAL ANALYSIS CHRISTIAN REMLING 1. Metric and topological spaces A metric space is a set on which we can measure distances. More precisely, we proceed as follows: let X 6= ; be a set, and let d : X×X ! [0; 1) be a map. Definition 1.1. (X; d) is called a metric space if d has the following properties, for arbitrary x; y; z 2 X: (1) d(x; y) = 0 () x = y (2) d(x; y) = d(y; x) (3) d(x; y) ≤ d(x; z) + d(z; y) Property 3 is called the triangle inequality. It says that a detour via z will not give a shortcut when going from x to y. The notion of a metric space is very flexible and general, and there are many different examples. We now compile a preliminary list of metric spaces. Example 1.1. If X 6= ; is an arbitrary set, then ( 0 x = y d(x; y) = 1 x 6= y defines a metric on X. Exercise 1.1. Check this. This example does not look particularly interesting, but it does sat- isfy the requirements from Definition 1.1. Example 1.2. X = C with the metric d(x; y) = jx−yj is a metric space. X can also be an arbitrary non-empty subset of C, for example X = R. In fact, this works in complete generality: If (X; d) is a metric space and Y ⊆ X, then Y with the same metric is a metric space also. Example 1.3. Let X = Cn or X = Rn. For each p ≥ 1, n !1=p X p dp(x; y) = jxj − yjj j=1 1 2 CHRISTIAN REMLING defines a metric on X. -
Ch. 15 Power Series, Taylor Series
Ch. 15 Power Series, Taylor Series 서울대학교 조선해양공학과 서유택 2017.12 ※ 본 강의 자료는 이규열, 장범선, 노명일 교수님께서 만드신 자료를 바탕으로 일부 편집한 것입니다. Seoul National 1 Univ. 15.1 Sequences (수열), Series (급수), Convergence Tests (수렴판정) Sequences: Obtained by assigning to each positive integer n a number zn z . Term: zn z1, z 2, or z 1, z 2 , or briefly zn N . Real sequence (실수열): Sequence whose terms are real Convergence . Convergent sequence (수렴수열): Sequence that has a limit c limznn c or simply z c n . For every ε > 0, we can find N such that Convergent complex sequence |zn c | for all n N → all terms zn with n > N lie in the open disk of radius ε and center c. Divergent sequence (발산수열): Sequence that does not converge. Seoul National 2 Univ. 15.1 Sequences, Series, Convergence Tests Convergence . Convergent sequence: Sequence that has a limit c Ex. 1 Convergent and Divergent Sequences iin 11 Sequence i , , , , is convergent with limit 0. n 2 3 4 limznn c or simply z c n Sequence i n i , 1, i, 1, is divergent. n Sequence {zn} with zn = (1 + i ) is divergent. Seoul National 3 Univ. 15.1 Sequences, Series, Convergence Tests Theorem 1 Sequences of the Real and the Imaginary Parts . A sequence z1, z2, z3, … of complex numbers zn = xn + iyn converges to c = a + ib . if and only if the sequence of the real parts x1, x2, … converges to a . and the sequence of the imaginary parts y1, y2, … converges to b. Ex. -
Topic 7 Notes 7 Taylor and Laurent Series
Topic 7 Notes Jeremy Orloff 7 Taylor and Laurent series 7.1 Introduction We originally defined an analytic function as one where the derivative, defined as a limit of ratios, existed. We went on to prove Cauchy's theorem and Cauchy's integral formula. These revealed some deep properties of analytic functions, e.g. the existence of derivatives of all orders. Our goal in this topic is to express analytic functions as infinite power series. This will lead us to Taylor series. When a complex function has an isolated singularity at a point we will replace Taylor series by Laurent series. Not surprisingly we will derive these series from Cauchy's integral formula. Although we come to power series representations after exploring other properties of analytic functions, they will be one of our main tools in understanding and computing with analytic functions. 7.2 Geometric series Having a detailed understanding of geometric series will enable us to use Cauchy's integral formula to understand power series representations of analytic functions. We start with the definition: Definition. A finite geometric series has one of the following (all equivalent) forms. 2 3 n Sn = a(1 + r + r + r + ::: + r ) = a + ar + ar2 + ar3 + ::: + arn n X = arj j=0 n X = a rj j=0 The number r is called the ratio of the geometric series because it is the ratio of consecutive terms of the series. Theorem. The sum of a finite geometric series is given by a(1 − rn+1) S = a(1 + r + r2 + r3 + ::: + rn) = : (1) n 1 − r Proof. -
11.3-11.4 Integral and Comparison Tests
11.3-11.4 Integral and Comparison Tests The Integral Test: Suppose a function f(x) is continuous, positive, and decreasing on [1; 1). Let an 1 P R 1 be defined by an = f(n). Then, the series an and the improper integral 1 f(x) dx either BOTH n=1 CONVERGE OR BOTH DIVERGE. Notes: • For the integral test, when we say that f must be decreasing, it is actually enough that f is EVENTUALLY ALWAYS DECREASING. In other words, as long as f is always decreasing after a certain point, the \decreasing" requirement is satisfied. • If the improper integral converges to a value A, this does NOT mean the sum of the series is A. Why? The integral of a function will give us all the area under a continuous curve, while the series is a sum of distinct, separate terms. • The index and interval do not always need to start with 1. Examples: Determine whether the following series converge or diverge. 1 n2 • X n2 + 9 n=1 1 2 • X n2 + 9 n=3 1 1 n • X n2 + 1 n=1 1 ln n • X n n=2 Z 1 1 p-series: We saw in Section 8.9 that the integral p dx converges if p > 1 and diverges if p ≤ 1. So, by 1 x 1 1 the Integral Test, the p-series X converges if p > 1 and diverges if p ≤ 1. np n=1 Notes: 1 1 • When p = 1, the series X is called the harmonic series. n n=1 • Any constant multiple of a convergent p-series is also convergent. -
1.1 Constructing the Real Numbers
18.095 Lecture Series in Mathematics IAP 2015 Lecture #1 01/05/2015 What is number theory? The study of numbers of course! But what is a number? • N = f0; 1; 2; 3;:::g can be defined in several ways: { Finite ordinals (0 := fg, n + 1 := n [ fng). { Finite cardinals (isomorphism classes of finite sets). { Strings over a unary alphabet (\", \1", \11", \111", . ). N is a commutative semiring: addition and multiplication satisfy the usual commu- tative/associative/distributive properties with identities 0 and 1 (and 0 annihilates). Totally ordered (as ordinals/cardinals), making it a (positive) ordered semiring. • Z = {±n : n 2 Ng.A commutative ring (commutative semiring, additive inverses). Contains Z>0 = N − f0g closed under +; × with Z = −Z>0 t f0g t Z>0. This makes Z an ordered ring (in fact, an ordered domain). • Q = fa=b : a; 2 Z; b 6= 0g= ∼, where a=b ∼ c=d if ad = bc. A field (commutative ring, multiplicative inverses, 0 6= 1) containing Z = fn=1g. Contains Q>0 = fa=b : a; b 2 Z>0g closed under +; × with Q = −Q>0 t f0g t Q>0. This makes Q an ordered field. • R is the completion of Q, making it a complete ordered field. Each of the algebraic structures N; Z; Q; R is canonical in the following sense: every non-trivial algebraic structure of the same type (ordered semiring, ordered ring, ordered field, complete ordered field) contains a copy of N; Z; Q; R inside it. 1.1 Constructing the real numbers What do we mean by the completion of Q? There are two possibilities: 1. -
Course 221: Michaelmas Term 2006 Section 3: Complete Metric Spaces, Normed Vector Spaces and Banach Spaces
Course 221: Michaelmas Term 2006 Section 3: Complete Metric Spaces, Normed Vector Spaces and Banach Spaces David R. Wilkins Copyright c David R. Wilkins 1997–2006 Contents 3 Complete Metric Spaces, Normed Vector Spaces and Banach Spaces 2 3.1 The Least Upper Bound Principle . 2 3.2 Monotonic Sequences of Real Numbers . 2 3.3 Upper and Lower Limits of Bounded Sequences of Real Numbers 3 3.4 Convergence of Sequences in Euclidean Space . 5 3.5 Cauchy’s Criterion for Convergence . 5 3.6 The Bolzano-Weierstrass Theorem . 7 3.7 Complete Metric Spaces . 9 3.8 Normed Vector Spaces . 11 3.9 Bounded Linear Transformations . 15 3.10 Spaces of Bounded Continuous Functions on a Metric Space . 19 3.11 The Contraction Mapping Theorem and Picard’s Theorem . 20 3.12 The Completion of a Metric Space . 23 1 3 Complete Metric Spaces, Normed Vector Spaces and Banach Spaces 3.1 The Least Upper Bound Principle A set S of real numbers is said to be bounded above if there exists some real number B such x ≤ B for all x ∈ S. Similarly a set S of real numbers is said to be bounded below if there exists some real number A such that x ≥ A for all x ∈ S. A set S of real numbers is said to be bounded if it is bounded above and below. Thus a set S of real numbers is bounded if and only if there exist real numbers A and B such that A ≤ x ≤ B for all x ∈ S. -
General Topology
General Topology Tom Leinster 2014{15 Contents A Topological spaces2 A1 Review of metric spaces.......................2 A2 The definition of topological space.................8 A3 Metrics versus topologies....................... 13 A4 Continuous maps........................... 17 A5 When are two spaces homeomorphic?................ 22 A6 Topological properties........................ 26 A7 Bases................................. 28 A8 Closure and interior......................... 31 A9 Subspaces (new spaces from old, 1)................. 35 A10 Products (new spaces from old, 2)................. 39 A11 Quotients (new spaces from old, 3)................. 43 A12 Review of ChapterA......................... 48 B Compactness 51 B1 The definition of compactness.................... 51 B2 Closed bounded intervals are compact............... 55 B3 Compactness and subspaces..................... 56 B4 Compactness and products..................... 58 B5 The compact subsets of Rn ..................... 59 B6 Compactness and quotients (and images)............. 61 B7 Compact metric spaces........................ 64 C Connectedness 68 C1 The definition of connectedness................... 68 C2 Connected subsets of the real line.................. 72 C3 Path-connectedness.......................... 76 C4 Connected-components and path-components........... 80 1 Chapter A Topological spaces A1 Review of metric spaces For the lecture of Thursday, 18 September 2014 Almost everything in this section should have been covered in Honours Analysis, with the possible exception of some of the examples. For that reason, this lecture is longer than usual. Definition A1.1 Let X be a set. A metric on X is a function d: X × X ! [0; 1) with the following three properties: • d(x; y) = 0 () x = y, for x; y 2 X; • d(x; y) + d(y; z) ≥ d(x; z) for all x; y; z 2 X (triangle inequality); • d(x; y) = d(y; x) for all x; y 2 X (symmetry). -
Be a Metric Space
2 The University of Sydney show that Z is closed in R. The complement of Z in R is the union of all the Pure Mathematics 3901 open intervals (n, n + 1), where n runs through all of Z, and this is open since every union of open sets is open. So Z is closed. Metric Spaces 2000 Alternatively, let (an) be a Cauchy sequence in Z. Choose an integer N such that d(xn, xm) < 1 for all n ≥ N. Put x = xN . Then for all n ≥ N we have Tutorial 5 |xn − x| = d(xn, xN ) < 1. But xn, x ∈ Z, and since two distinct integers always differ by at least 1 it follows that xn = x. This holds for all n > N. 1. Let X = (X, d) be a metric space. Let (xn) and (yn) be two sequences in X So xn → x as n → ∞ (since for all ε > 0 we have 0 = d(xn, x) < ε for all such that (yn) is a Cauchy sequence and d(xn, yn) → 0 as n → ∞. Prove that n > N). (i)(xn) is a Cauchy sequence in X, and 4. (i) Show that if D is a metric on the set X and f: Y → X is an injective (ii)(xn) converges to a limit x if and only if (yn) also converges to x. function then the formula d(a, b) = D(f(a), f(b)) defines a metric d on Y , and use this to show that d(m, n) = |m−1 − n−1| defines a metric Solution. -
DEFINITIONS and THEOREMS in GENERAL TOPOLOGY 1. Basic
DEFINITIONS AND THEOREMS IN GENERAL TOPOLOGY 1. Basic definitions. A topology on a set X is defined by a family O of subsets of X, the open sets of the topology, satisfying the axioms: (i) ; and X are in O; (ii) the intersection of finitely many sets in O is in O; (iii) arbitrary unions of sets in O are in O. Alternatively, a topology may be defined by the neighborhoods U(p) of an arbitrary point p 2 X, where p 2 U(p) and, in addition: (i) If U1;U2 are neighborhoods of p, there exists U3 neighborhood of p, such that U3 ⊂ U1 \ U2; (ii) If U is a neighborhood of p and q 2 U, there exists a neighborhood V of q so that V ⊂ U. A topology is Hausdorff if any distinct points p 6= q admit disjoint neigh- borhoods. This is almost always assumed. A set C ⊂ X is closed if its complement is open. The closure A¯ of a set A ⊂ X is the intersection of all closed sets containing X. A subset A ⊂ X is dense in X if A¯ = X. A point x 2 X is a cluster point of a subset A ⊂ X if any neighborhood of x contains a point of A distinct from x. If A0 denotes the set of cluster points, then A¯ = A [ A0: A map f : X ! Y of topological spaces is continuous at p 2 X if for any open neighborhood V ⊂ Y of f(p), there exists an open neighborhood U ⊂ X of p so that f(U) ⊂ V . -
Formal Power Series Rings, Inverse Limits, and I-Adic Completions of Rings
Formal power series rings, inverse limits, and I-adic completions of rings Formal semigroup rings and formal power series rings We next want to explore the notion of a (formal) power series ring in finitely many variables over a ring R, and show that it is Noetherian when R is. But we begin with a definition in much greater generality. Let S be a commutative semigroup (which will have identity 1S = 1) written multi- plicatively. The semigroup ring of S with coefficients in R may be thought of as the free R-module with basis S, with multiplication defined by the rule h k X X 0 0 X X 0 ( risi)( rjsj) = ( rirj)s: i=1 j=1 s2S 0 sisj =s We next want to construct a much larger ring in which infinite sums of multiples of elements of S are allowed. In order to insure that multiplication is well-defined, from now on we assume that S has the following additional property: (#) For all s 2 S, f(s1; s2) 2 S × S : s1s2 = sg is finite. Thus, each element of S has only finitely many factorizations as a product of two k1 kn elements. For example, we may take S to be the set of all monomials fx1 ··· xn : n (k1; : : : ; kn) 2 N g in n variables. For this chocie of S, the usual semigroup ring R[S] may be identified with the polynomial ring R[x1; : : : ; xn] in n indeterminates over R. We next construct a formal semigroup ring denoted R[[S]]: we may think of this ring formally as consisting of all functions from S to R, but we shall indicate elements of the P ring notationally as (possibly infinite) formal sums s2S rss, where the function corre- sponding to this formal sum maps s to rs for all s 2 S. -
Absolute Convergence in Ordered Fields
Absolute Convergence in Ordered Fields Kristine Hampton Abstract This paper reviews Absolute Convergence in Ordered Fields by Clark and Diepeveen [1]. Contents 1 Introduction 2 2 Definition of Terms 2 2.1 Field . 2 2.2 Ordered Field . 3 2.3 The Symbol .......................... 3 2.4 Sequential Completeness and Cauchy Sequences . 3 2.5 New Sequences . 4 3 Summary of Results 4 4 Proof of Main Theorem 5 4.1 Main Theorem . 5 4.2 Proof . 6 5 Further Applications 8 5.1 Infinite Series in Ordered Fields . 8 5.2 Implications for Convergence Tests . 9 5.3 Uniform Convergence in Ordered Fields . 9 5.4 Integral Convergence in Ordered Fields . 9 1 1 Introduction In Absolute Convergence in Ordered Fields [1], the authors attempt to dis- tinguish between convergence and absolute convergence in ordered fields. In particular, Archimedean and non-Archimedean fields (to be defined later) are examined. In each field, the possibilities for absolute convergence with and without convergence are considered. Ultimately, the paper [1] attempts to offer conditions on various fields that guarantee if a series is convergent or not in that field if it is absolutely convergent. The results end up exposing a reliance upon sequential completeness in a field for any statements on the behavior of convergence in relation to absolute convergence to be made, and vice versa. The paper makes a noted attempt to be readable for a variety of mathematic levels, explaining new topics and ideas that might be confusing along the way. To understand the paper, only a basic understanding of series and convergence in R is required, although having a basic understanding of ordered fields would be ideal. -
The Ratio Test, Integral Test, and Absolute Convergence
MATH 1D, WEEK 3 { THE RATIO TEST, INTEGRAL TEST, AND ABSOLUTE CONVERGENCE INSTRUCTOR: PADRAIC BARTLETT Abstract. These are the lecture notes from week 3 of Ma1d, the Caltech mathematics course on sequences and series. 1. Homework 1 data • HW average: 91%. • Comments: none in particular { people seemed to be pretty capable with this material. 2. The Ratio Test So: thus far, we've developed a few tools for determining the convergence or divergence of series. Specifically, the main tools we developed last week were the two comparison tests, which gave us a number of tools for determining whether a series converged by comparing it to other, known series. However, sometimes this P 1 P n isn't enough; simply comparing things to n and r usually can only get us so far. Hence, the creation of the following test: Theorem 2.1. If fang is a sequence of positive numbers such that a lim n+1 = r; n!1 an for some real number r, then P1 • n=1 an converges if r < 1, while P1 • n=1 an diverges if r > 1. If r = 1, this test is inconclusive and tells us nothing. Proof. The basic idea motivating this theorem is the following: if the ratios an+1 an eventually approach some value r, then this series eventually \looks like" the geo- metric series P rn; consequently, it should converge whenever r < 1 and diverge whenever r > 1. an+1 To make the above concrete: suppose first that r > 1. Then, because limn!1 = an r;, we know that there is some N 2 N such that 8n ≥ N, a n+1 > 1 an )an+1 > an: But this means that the an's are eventually an increasing sequence starting at some value aN > 0; thus, we have that limn!1 an 6= 0 and thus that the sum P1 n=1 an cannot converge.