Buck Meta Chapter 1 Sets and Functions

Buck Meta Chapter 1 Notes PhL 1.11.15 Meta Notes. I took this approach with Stakgold. While the original raw notes contain lots of detail and added comments, examples, questions and their answers, the Meta notes are at a higher level and try to give a simple and compact summary of the content. Chapter 1: Sets and Functions 1 1.1 A short intro. 1 1.2 The Geometry of En Cartesian space.[2] 1 1.3 Distance and Norms. [9] // Theorem 1 1 1.4 Sets and Functions. [15] // no theorems 1 1.5 Topological ideas. [24] // Theorem 2 1 1.6 Ordering for R ≡ E1 and grab bag of other topics [34]. // Theorems 3→5 BW + HB 2 1.7 Sequences [40] // Theorems 6→11 2 1.8 Real sequences [48] // Theorems 12,13 3 Chapter 1: Sets and Functions [ pp 1-54 ] 1.1 A short intro. 1.2 The Geometry of En Cartesian space.[2] Algebraic refers so the field properties of En = n-space. Vector space. Quadrilateral midpoints always mark a parallelogram. 1.3 Distance and Norms. [9] // Theorem 1 Distance, Norms, absolute value, inner products. Theorem 1: Schwarz Inequality proved: pq ≤ |p| |q| (involves norms and inner product) Triangle Inequality: [12] Says that |p+q| ≤ |p| + |q| ( proved from Schwarz, hence corollary of) Straight Line as p + λ(p-q) = p(1-λ) + λq, segment p to q is then 0 ≤ λ ≤ 1 . Convexity of a set in En, how related to above line segment idea. 1.4 Sets and Functions. [15] // no theorems Contains and covers means same, set and collection means same. Set basics: union, intersection, notation and so on. Functions, domain of, function as mapping, curves and surfaces. Sequence as a mapping n→an or integers to some space. Trace is range/image of sequence. Notion of A x B as the graph of f: A→B, is called Cartesian Product 1.5 Topological ideas. [24] // Theorem 2 Neighborhood (ball) of a point in a set. Interior point has neighborhood entirely in set. Exterior point has neighborhood entirely outside set. Boundary point is neither in or out, boundary denoted ∂S. Open set has only interior points. Closed set means all points not in set are exterior points, or ∂S is in S. Closure of set S is S ∂S = is a closed set, but Bucks don't use that overbar notation. Complement of an open set is closed and vice versa. (a fact) Cluster point = accumulation point = limit point: any ball has ∞ points in S. Isolated point = can have ball with no other points, not a cluster point Facts about open and closed sets page 28. Connected = cannot split S into two non-null disjoint sets with no shared ∂S elements Countable and non-countable sets [30]. Bounded and unbounded sets (requires a norm for a point in the set, |p| ) Topology on set S = a description of the open subsets of S U is open relative to S means U is intersection of an open set in E2 with set S, p 32 fig. Theorem 2. Set S = not connected S = AB where A and B are disjoint and are each open relative to S. [ page 32 ] 1.6 Ordering for R ≡ E1 and grab bag of other topics [34]. // Theorems 3→5 BW + HB Order relation < in sense that a < b for a,b in E1. Notation [a,b] for closed interval, and how interval is specified using <. Symbol ∞ (page 35) LUB and GLB of a set with an order relation < Theorem 3. The line is a connected set. A dense in B means = B. Rationals are dense in the reals. Rationals called R0 and form a disconnected set (break or gap at every non-rational number) Set is complete if there are no such gaps. Nested Set Theorem [37]: for an infinite nested set where all Ci are closed and bounded, there is a point which is in all the sets. Theorem 4. (Bolzano-Weierstrass). Every bounded infinite set S in En has at least one cluster point. Heine-Borel Property: A set has this property if any cover of open sets has a finite subcover. A set with this property is said to be compact. Theorem 5. (Heine-Borel). Any closed and bounded set in En has the Heine-Borel property (is compact). 1.7 Sequences [40] // Theorems 6→11 Definition of a sequence of points {pn}: mapping from I → En. Note that a sequence means an infinite sequence in this book. Sequence converges to p ≡ any neighborhood of p has all points pn from some N on. {pn} is divergent if there is no p to which it converges (does not mean pn→∞) Theorem 6: A convergent sequence is bounded. This seems pretty obvious and proof is tiny. Theorem 7: Sum of two convergent sequences converges to the obvious sum point. Theorem 8: The set consists of points which are all limit points of convergent sequences. Subsequence definition [44] Theorem 9. Any bounded sequence has a convergent subsequence. [ Example p 41] A Convergent (to p) Sequence has | pk - p | < any ε for k being large enough. A Cauchy Sequence has | pk - pj | < any ε for both k and j being large enough. Fact: Every convergent sequence is a Cauchy sequence. [ p 45, Exercise 11] Theorem 10. Every Cauchy sequence is a convergent sequence (in En). Theorem 11. If | pj+1 - pj | < cj where |c| < 1, then {pj} converges. Theorem of Exercise 6: If {pn} has a limit point p, then {pn} has a subsequence which converges to that limit point (perhaps the full sequence jumps around if non-convergent as on page 41). 1.8 Real sequences [48] // Theorems 12,13 Monotonic = can be up or down [48] Theorem 12: In R, a bounded monotonic sequence must converge. Theorem 13: For any real x, there is always an integer n such that n > x. Notion of sequence convergence from above or from below in R. Use of lim sup an and lim inf an to handle sequences that jump around and have therefore multiple cluster points and do not converge. These can exist when lim an does not exist. The sup and inf are the largest and smallest limit points of a sequence. Notion of rate of convergence of a sequence. Note: The subject of series and sequences of functions has not yet been brought up.