Metric Spaces DEF. A metric space X or (X; d) is a nonempty set X together with a function d : X X! [0; 1) such that for all x; y; and z in X : 1. d (x; y) 0 with equality i x = y 2. d (x; y) = d (y; x) 3. d (x; z) d (x; y) + d (y; z) DEF. If (X; d) is a metric space and E is a nonempty subset, then (E; d) is also a metric space, called a subspace of X: DEF. Let (X; d) be a metric space, x 2 X; and r > 0: 1. The open ball with center x and radius r is B r (x) = fy : d (y; x) < rg 2. The closed ball with center x and radius r is B r (x) = fy : d (y; x) rg DEF. A subset E in a metric space X is open if for every point x in E there is an r > 0 such that the B r (x) E: DEF. A subset E in a metric space X is closed if its complement, E c = XnE is open. Facts: 1. Open balls are open sets. 2. Closed balls are closed sets. DEF. A point x of a subset E in a metric space X is an interior point of E if there is an r > 0 such that B r (x) E: The interior of E; denoted by E is the set of all interior points of E: DEF. A point x in a metric space X is a closure point (point of closure) of a subset E of X if B " (x) \ E 6= ; for all " > 0: The closure of E; denoted by E is the set of all closure points of E: Facts: 1
1. E is an open set. 2. E E: 3. A set E is open i all its points are interior points; equivalently, i E = E. 4. E is an closed set. 5. E E: 6. A set E is closed i E contains all its closure points; equivalently, E = E. DEF. Let x be a point in a metric space X: A neighborhood of x is any open set V in X that contains x: DEF. A subset E in a metric space X is bounded if there is a point a 2 X and a constant M > 0 such that d (x; a) < M for all x 2 E: Fact: If a set E in a metric space is bounded then given any b 2 X there is a constant R > 0 such that d (x; b) < R for all x 2 E: DEF. A subset D in a metric space X is dense in X if for every point x 2 X and every neighborhood V of x it is true that D \ V 6= ;: Sequences in Metric Spaces DEF. A sequence fx n g 1 n=1 whose value at n is x n : in a metric space X is a function from N into X DEF. A sequence fx n g 1 n=1 in a metric space X: 1. fx n g converges if there is a point x 2 X, called its limit, such that for every " > 0 there exists an n 2 N such that n N =) d (x n ; x) < "; equivalently, d (x n ; x)! 0 as n! 1: 2. fx n g is Cauchy (a Cauchy sequence) if for every " > 0 there exists an n 2 N such that n; m N =) d (x n ; x m ) < ": 2
3. fx n g is bounded if there exists an M > 0 and a point a 2 X such that d (x n ; a) < M for all n 2 N. Facts: (for CDP and proved essentially by replacing jx corresponding proofs for sequences in R:) yj by d (x; y) in the 1. A sequence has at most one limit. 2. If a sequence converges, all its subsequences converge and the subsequences have the same limit as the sequence. 3. Every convergence sequence is bounded. 4. Every Cauchy sequence is bounded. 5. Every convergent sequence is Cauchy. 6. If fx n g is bounded, then given any b 2 X there is an M > 0 such that d (x n ; b) < M for all n 2 N. Th. (HW #3.) Let x 2 (X; d) : A sequence fx n g converges to x i for every neighborhood V of x there exist n 2 N such that n N implies x n 2 V: Th. Let x 2 (X; d) : A sequence fx n g converges to x i for every neighborhood V of x there exist n 2 N such that n N implies x n 2 V: Th. A point x 2 (X; d) is a closure point of a subset E X i there is a sequence fx n g in E with limit x: Cor. A set E (X; d) is closed i every sequence fx n g in E that converges (in X) has its limit in E: The Corollary is for CDP. DEF. A metric space X is complete if every Cauchy sequence (in X) converges (in X). Fact: Not all metric spaces are complete. 3
Normed Linear Spaces (Metric and linear structure combined) DEF. Linear space is a synonym for vector space. DEF. Let V be a vector space over R. A norm on V is a function kk : V! R with the following properties i. kxk 0 with equality only if x = 0 ii. kxk = jj kxk iii. kx + yk kxk + kyk for all x; y 2 V and all scalars 2 R. DEF. A (real) normed linear space is a vector space over R together with a norm de ned on it. If V is the vector space and kk its norm, the normed space is denoted by either V or (V; kk) depending on the context. DEF. d (x; y) = kx (HW #2.) yk de nes a metric on V is called the induced metric. Whenever we refer to a metric property of V; we mean that V is viewed as a metric space with the induced metric (unless explicit mention to the contrary is made). CDP: If V is a normed linear space: 1. Express the de nition of convergence if fx n g directly in term of the norm. 2. Express the de nition of Cauchy sequence directly in term of the norm. 3. Express the de nition of bounded set (and bounded sequence) directly in term of the norm using balls centered at the origin. Th. Let V be a normed linear space. If fx n g and fy n g are convergent sequences in V and is any scalar, then Furthermore, as n! 1: 9 lim n + y n ) n!1 = lim n + lim n!1 9 lim n) n!1 = lim n n!1 x n! x =) kx n k! kxk n!1 y n Th. (HW #2) If R n is equipped with either the 1-norm, 2-norm, or 1-norm, then a sequence x (k)! x in R n as k! 1 if and only if each component of x (k) converges to the corresponding component of x: That is, if x (k) = x (k) 1 ; x(k) 2 ; :::; x(k) n and x = (x 1 ; x 2 ; :::; x n ) 4
then Here (k) is a superscript. x (k) j! x j for each j = 1; 2; :::; n: Cor. In R n with the usual norm, if x (k)! x and y (k)! y as k! 1; then as k! 1: x (k) y (k)! x y Th. R n with the usual norm (usual metric) is complete; that is, every Cauchy sequence in R n converges. Th. (Bolzano-Weierstrass in R n ) Every bounded sequence in R n has a convergent subsequence. Th. (HW #4) Q n ; the set of points in R n all of whose cooridnates are rational, is dense in R n. Limits of Functions Let X = [0; 1][f2g with the metric inherited from R. Is it sensible to inquire about lim x!2 f (x) for some function f (x)? 1. DEF. Let X be a metric space and E a subset (which can be X): A point a 2 X is a cluster point (AKA accumulation point) of E if every neighborhood of a contains in nitely many points of E: 2. Facts (CDP): 1. a is a cluster point of E i every neighborhood of E contains at least two points of E: 2. If a is a cluster point of E; then a is a closure point of E; but not conversely. 5
3. DEF. Let X and Y be a metric spaces, a 2 X be a cluster point of X and L 2 Y: Let f : Xn fag! Y: Then lim f (x) = L x!a means that for every " > 0 there exists a > 0 such that 0 < d (x; a) < =) d (f (x) ; L) < ": lim f (x) x!a = L is also expressed as f (x)! L as x! a: The following results can be proven by taking the proof you learned in MTH 311 and replacing distances expressed in terms of absolute values in R by the corresponding distances in the metric spaces X and Y: 4. Th. Let X and Y be a metric spaces, a 2 X be a cluster point of X and f; g : Xn fag! Y, and L 2 Y: Then 1. If f (x) = g (x) on Xn fag ; then 9 lim x!a f (x) () 9 lim x!a g (x) in which case both limits are equal. 2. 9 lim x!a f (x) = L () for every sequence fx n g in Xn fag with limit a, it follows that f (x n )! L as n! 1: When the range space of the functions is a vector space we get algebraic limit laws, to the extent we could reasonably expect to have them: 5. Th. Let X be a metric space and a 2 X be a cluster point of X: Let Y be a normed linear space and f; g : Xn fag! Y and be a scalar. Then if f and g both have limits as x! a; the following limits exist and have the indicated values: 1. lim x!a (f + g) (x) = lim x!a f (x) + lim x!a g (x) 2. lim x!a (f) (x) = lim x!a f (x) 6. Th. Let X be a metric space and a 2 X be a cluster point of X: Let f; g : Xn fag! R n. If f and g both limits as x! a; the following limit exists and has the indicated value: lim (f g) (x) = lim f (x) lim g (x) : x!a x!a x!a 6
When Y = R we get more algebraic limit laws, squeeze laws, and comparison laws: 7. Th. Let X be a metric space and a 2 X be a cluster point of X: Let f; g : Xn fag! R and be a scalar. Then if f and g both have limits as x! a; the following limits exist and have the indicated values: 1. lim x!a (fg) (x) = lim x!a f (x) lim x!a f (x) : 2. lim x!a f g (x) = lim x!a f(x) g(x) provided lim x!a g (x) 6= 0: 8. Th. (Squeeze Law) Let X be a metric space and a 2 X be a cluster point of X: Let f; g; and h be real-valued functions whose domains include B r (a) n fag for some r > 0 and satisfy f (x) h (x) g (x) there. If f and g both have limits as x! a and these limits are equal, then 9 lim x!a h (x) = lim x!a f (x) = lim x!a g (x) : 9. Th. (Comparison Law) Let X be a metric space and a 2 X be a cluster point of X: Let f; and g be real-valued functions whose domains include B r (a) n fag for some r > 0 and satisfy f (x) g (x) there. If f and g both have limits as x! a; then lim f (x) lim g (x) : x!a x!a Continuity 10. DEF. Let E be a nonempty subset of a metric space X and Y be a metric space. A function f : E! Y is continuous at a 2 E if to each " > 0 there corresponds a > 0 such that x 2 E and d (x; a) < =) d (f (x) ; f (a)) < ": We say f is continuous on E if it is continuous at every point of E: We say f is continuous if it is continuous on its domain. The expected properties of continuity hold and follow directly from our results about limits of functions. The proofs are the same as for the corresponding results in MTH 311. 11. Th. Let X and Y be a metric spaces, E X; and f; g : E! Y: Then 7
1. f is continuous at a 2 E if and only if for every sequence fx n g in E with limit a, it follows that f (x n )! f (a) as n! 1: 2. If Y is a normed linear space and f and g are continuous at a, then f + g is continuous at a for all scalars and ; 3. If, in addition, Y = R n, then f g is continuous at a and in the special case of n = 1; f=g is continuous at a provided g (a) 6= 0: 12. Th. (The Composition of Continuous Functions is Continuous) Let X; Y; and Z be a metric spaces, E X; f : E! Y; and g : Y! Z: If f is continuous at a 2 E and g is continuous at b = f (a), then g f is continuous at a: 13. DEF. Let E be a nonempty subset of a metric space X and Y be a metric space. A function f : E! Y is uniformly continuous on E if to each " > 0 there corresponds a > 0 such that 14. Facts: x; x 0 2 E and d (x; x 0 ) < =) d (f (x) ; f (x 0 )) < ": 1. f uniformly continuous on E implies f continuous on E: 2. f continuous on E need not imply f uniformly continuous on E: Sequential Compactness Compactness is a proxy for niteness when niteness is not present. For example, if a function has domain an nite set, then it obviously assumes its maximum and minimum values. We know the same is true for a real-valued function on a closed bounded interval [a; b] : The interval is compact in the sense we are now going to de ne. 15. DEF. A subset E of a metric space X is sequentially compact if every sequence in E has a convergent subsequence that converges to a point of E: 16. Facts: 1. There is a more general notion than sequential compactness called compactness. In a metric space these two properties are equivalent. So from now on I will shorten sequentially compact to just compact. 2. Every nite set E in a metric space is compact. 3. A closed bounded interval [a; b] in R is compact. 8
17. Th. A compact set in any metric space is closed and bounded. 18. Th. (Heine-Borel) A subset of R n is compact if and only if it is closed and bounded. 19. Th. (Max-Min Theorem) Let f : E! R be a continuous function on a compact subset E of a metric space X: Then f assumes maximum and minimum values on X: That is, there exist points and in E such that f () f (x) f () for all x 2 E: 20. Th. All norms on R n are equivalent. 21. Th. Let E be a compact subset of a metric space X; and Y be a metric space. If f : E! Y is continuous on E; then f is uniformly continuous on E: Connectedness 22. DEF. A subset E in metric space X is connected if whenever E = (E \ U) [ (E \ V ) where U and V are disjoint (U \ V = ;) open sets in X, then either E \ U = ; or E \ V = ; Intuitively E connected means that E cannot be split into two separate pieces in X. 23. Remark: If the subset E = X in the de nition of connected set, the de nition says a metric space X is connected i whenever X = U [ V where U and V are disjoint (U \ V = ;) open sets in X, then either U = ; or V = ; 24. Facts: 9
1. E connected implies E is connected. 2. E and F connected and E \ F 6= ; implies E [ F connected. 3. E R connected () E is an interval (of some type). 25. Facts About Metric Subspaces (CDP): Let E be a nonempty subset of a metric space (X; d) : Then E is a metric space in its own right, (E; d) ; with the induced metric d (which means d restricted to E E). 1. Let x 2 E and r > 0: Then B E r (x) = E \ B r (x) : 2. U is open in the metric space (E; d) if and only if there is an open set ~ U in X such that U = E \ ~ U: 3. C is closed in the metric space (E; d) if and only if there is a closed set ~ C in X such that C = E \ ~ C: The de nition of connected subset and the foregoing facts yield the following result. 25. Th. A nonempty subset E of a metric space (X; d) is connected if and only if the metric space (E; d) is connected. 26. Th. (Intermediate Value Theorem) If f : E (X; d)! R is continuous and E is connected, then the range of f is an interval. Uniform Limits of Continuous Functions 27. Let X be a metric space and f n : X! R be a sequence of continuous functions. If f n converges uniformly on X to a functon f : X! R, then f is continuous on X: DEF. Let X and Y be a metric spaces. The set of all continuous functions from X to Y is denoted by C (X; Y ) which may be abbreviated to C (X) or to C: 28. Th. If X is a compact metric space, then C (X; R) is a complete normed linear space with the max norm (AKA the 1-norm). Cor. C ([a; b] ; R) is a complete normed linear space with the max norm. Cor. C (E; R) is a complete normed linear space with the max norm for any closed bounded set E in R n : 10