MATH 3210 Metric spaces University of Leeds, School of Mathematics November 29, 2017 Syllabus: 1. Definition and fundamental properties of a metric space. Open sets, closed sets, closure and interior. Convergence of sequences. Continuity of mappings. (6) 2. Real inner-product spaces, orthonormal sequences, perpendicular distance to a subspace, applications in approximation theory. (7) 3. Cauchy sequences, completeness of R with the standard metric; uniform convergence and completeness of C[a; b] with the uniform metric. (3) 4. The contraction mapping theorem, with applications in the solution of equations and differential equations. (5) 5. Connectedness and path-connectedness. Introduction to compactness and sequential compactness, including subsets of Rn. (6) LECTURE 1 Books: Victor Bryant, Metric spaces: iteration and application, Cambridge, 1985. M. O.´ Searc´oid,Metric Spaces, Springer Undergraduate Mathematics Series, 2006. D. Kreider, An introduction to linear analysis, Addison-Wesley, 1966. 1 Metrics, open and closed sets We want to generalise the idea of distance between two points in the real line, given by d(x; y) = jx − yj; and the distance between two points in the plane, given by p 2 2 d(x; y) = d((x1; x2); (y1; y2)) = (x1 − y1) + (x2 − y2) : to other settings. [DIAGRAM] This will include the ideas of distances between functions, for example. 1 1.1 Definition Let X be a non-empty set. A metric on X, or distance function, associates to each pair of elements x, y 2 X a real number d(x; y) such that (i) d(x; y) ≥ 0; and d(x; y) = 0 () x = y (positive definite); (ii) d(x; y) = d(y; x) (symmetric); (iii) d(x; z) ≤ d(x; y) + d(y; z) (triangle inequality). Examples: (i) X = R. The standard metric is given by d(x; y) = jx − yj. There are many other metrics on R, for example d(x; y) = jex − eyj; jx − yj if jx − yj ≤ 1, d(x; y) = 1 if jx − yj ≥ 1. Let X be any set whatsoever, then we can define 1 if x 6= y, d(x; y) = (the discrete metric). 0 if x = y, 2 (ii) X = R . The standard metric is the Euclidean metric: if x = (x1; x2) and y = (y1; y2) then p 2 2 d2(x; y) = (x1 − y1) + (x2 − y2) : This is linked to the inner-product (scalar product), x:y = x1y1 + x2y2, since it is just p(x − y):(x − y). We will study inner products more carefully later, so for the moment we won't prove the (well-known) fact that it is indeed a metric. Other possible metrics include d1(x; y) = maxfjx1 − y1j; jx2 − y2jg: Let's check the axioms. In fact (i) and (ii) are easy (i.e., the distance is positive definite, symmetric); for (iii) let's write jx1 −y1j = p, jx2 −y2j = q, jy1 −z1j = r and jy2 −z2j = s. Then jx1 − z1j ≤ p + r and jx2 − z2j ≤ q + s; so d1(x; z) = maxfjx1 − z1j; jx2 − z2jg ≤ maxfp + r; q + sg ≤ maxfp; qg + maxfr; sg = d1(x; y) + d1(y; z): by inspection. 2 Another metric on R comes from d1(x; y) = jx1 − y1j + jx2 − y2j. These metrics are all translation-invariant (i.e., d(x + z; y + z) = d(x; y)), and homogeneous (i.e., d(kx; ky) = jkjd(x; y)). 2 (iii) Take X = C[a; b]. Here are three metrics: s Z b 2 d2(f; g) = (f(x) − g(x)) dx: a Again, this is linked to the idea of an inner product, so we will delay proving that it is a metric. Z b d1(f; g) = jf(x) − g(x)j dx; a the area between two curves [DIAGRAM]. d1(f; g) = maxfjf(x) − g(x)j : a ≤ x ≤ bg; the maximum separation between two curves. [DIAGRAM]. Example: on C[0; 1] take f(x) = x and g(x) = x2 and calculate Z 1 1=2 2 2 p d2(f; g) = (x − x ) dx = 1=30; 0 Z 1 2 d1(f; g) = jx − x j dx = 1=6; and 0 2 d1(f; g) = max jx − x j = 1=4: x2[0;1] 1.2 Definition A set X together with a metric d is called a metric space, sometimes written (X; d). If A ⊆ X then we can use d to measure distances between points of A, and (A; d) is also a metric space, called a subspace of (X; d). LECTURE 2 Examples: 1. The interval [a; b] with d(x; y) = jx − yj is a subspace of R. 2 2 2 p 2 2 2. The unit circle f(x1; x2) 2 R : x1 +x2 = 1g with d(x; y) = (x1 − y1) + (x2 − y2) is a subspace of R2. 3. The space of polynomials P is a metric space with any of the metrics inherited from C[a; b] above. 1.3 Definition 3 Let (X; d) be a metric space, let x 2 X and let r > 0. The open ball centred at x, with radius r, is the set B(x; r) = fy 2 X : d(x; y) < rg; and the closed ball is the set B[x; r] = fy 2 X : d(x; y) ≤ rg: Note that in R with the usual metric the open ball is B(x; r) = (x − r; x + r), an open interval, and the closed ball is B[x; r] = [x − r; x + r], a closed interval. 2 For the d2 metric on R , the unit ball, B(0; 1), is disc centred at the origin, excluding the boundary. You may like to think about what you get for other metrics on R2. 1.4 Definition A subset U of a metric space (X; d) is said to be open, if for each point x 2 U there is an r > 0 such that the open ball B(x; r) is contained in U (\room to swing a cat"). Clearly X itself is an open set, and by convention the empty set ; is also considered to be open. 1.5 Proposition Every \open ball" B(x; r) is an open set. Proof: For if y 2 B(x; r), choose δ = r − d(x; y). We claim that B(y; δ) ⊂ B(x; r). If z 2 B(y; δ), i.e., d(z; y) < δ, then by the triangle inequality d(z; x) ≤ d(z; y) + d(y; x) < δ + d(x; y) = r: So z 2 B(x; r). 1.6 Definition A subset F of (X; d) is said to be closed, if its complement X n F is open. Note that closed does not mean \not open". In a metric space the sets ; and X are both open and closed. In R we have: (a; b) is open. [a; b] is closed, since its complement (−∞; a) [ (b; 1) is open. [a; b) is not open, since there is no open ball B(a; r) contained in the set. Nor is it closed, since its complement (−∞; a) [ [b; 1) isn't open (no ball centred at b can be contained in the set). 1.7 Example 4 If we take the discrete metric, 1 if x 6= y, d(x; y) = 0 if x = y, then each point fxg = B(x; 1=2) so is an open set. Hence every set U is open, since for x 2 U we have B(x; 1=2) ⊆ U. Hence, by taking complements, every set is also closed. 1.8 Proposition In a metric space, every one-point set fx0g is closed. Proof: We need to show that the set U = fx 2 X : x 6= x0g is open, so take a point x 2 U. Now d(x; x0) > 0, and the ball B(x; r) is contained in U for every 0 < r < d(x; x0). [DIAGRAM] 1.9 Theorem Let (Uα)α2A be any collection of open subsets of a metric space (X; d) (not necessarily S finite!). Then α2A Uα is open. Let U and V be open subsets of a metric space (X; d). Then U\V is open. Hence (by induction) any finite intersection of open subsets is open. S Proof: If x 2 Uα then there is an α with x 2 Uα. Now Uα is open, so α2A S B(x; r) ⊂ Uα for some r > 0. Then B(x; r) ⊂ α2A Uα so the union is open. If now U and V are open and x 2 U \ V , then 9r > 0 and s > 0 such that B(x; r) ⊂ U and B(x; s) ⊂ V , since U and V are open. Then B(x; t) ⊂ U \ V if t ≤ min(r; s). [DIAGRAM.] So the collection of open sets is preserved by arbitrary unions and finite intersections. 1 1 However, an arbitrary intersection of open sets is not always open; for example (− n ; n ) T1 1 1 is open for each n = 1; 2; 3;:::, but n=1(− n ; n ) = f0g, which is not an open set. LECTURE 3 For closed sets we swap union and intersection. 1.10 Theorem Let (Fα)α2A be any collection of closed subsets of a metric space (X; d) (not necessar- T ily finite!). Then α2A Fα is closed. Let F and G be closed subsets of a metric space (X; d). Then F [ G is closed. Hence (by induction) any finite intersection of closed 5 subsets is closed. To prove this we recall de Morgan's laws. We use the notation Sc for the complement X n S of a set S ⊂ X. [ [ c \ c x 62 Aα () x 62 Aα for all α; so ( Aα) = Aα: α \ \ c [ c x 62 Aα () x 62 Aα for some α; so ( Aα) = Aα: α c S Proof: Write Uα = Fα = X n Fα which is open.