Analysis 3 Notes Namdar Homayounfar December 12, 2012 Contents 1 Definitions 2 2 Theorems, Lemmas, Propositions 12 1 1 Definitions Metric Space: A metric space is an ordered pair (X; d) where X is a set and d : X × X ! R+ a distance such that 1. d(x; y) ≥ 0 , d(x; y) = 0 () x = y 2. d(x; y) = d(y; x) 3. Triangle Inequality: d(x; y) ≤ d(x; z) + d(z; y) Norm: Let X be a vector space over R or C.A norm is a function k:k : X ! R such that 1. kxk ≥ 0 2. kxk = 0 () x = 0X 3. kαxk = jαj: kxk 4. kx + yk ≤ kxk + kyk Pn p Definition: `p = fx = (x1; :::; xn;::: ) j k=1 jxkj < 1g Definition: `1 = f bounded sequences g = fx = (x1; :::; xn;::: ) j kxk1 = supk jxkj < 1g ***** Topological Space: A topological space is a set X together with a collection O of subsets of X, called open sets, such that : S 1. The union of any collection of sets in O is in O. i.e. If fUαgα2I ⊂ O then α2I Uα 2 O. 2. The intersection of any finite collection of sets in O is in O. i.e. If U1;:::;Un are in O, then so is U1 \···\ Un 3. Both ; and X are in O. Some other terminologies: • The collection O of open sets is called a topology on X. • The collection O of all subsets of X, defines a topology on X called the discrete topology. • The set O = f;;Xg defines a topology, the trivial topology. • If O and O0 define two topologies on X, with every set that is open in O topology is also open in the O0 topology, i.e. O ⊂ O0, we say that O0 is finer than O an that O is coarser than O0. 2 Closed Set: A subset A of topological space X is closed if its complement XnA is open. ***** Let X be a topological space, A ⊂ X and x 2 X. Then only one of the following is true for x: 1. 9 an open U in X s.t. x 2 U ⊂ A 2. 9 an open V in X s.t. x 2 V ⊂ XnA 3. None of the above. 8U open such that x 2 U, we have U \ A 6= ; and U \ XnA 6 ; Interior of A := fx 2 X j (1) holdsg = A◦ Interior of XnA := fx 2 X j (2) holdsg Boundary of A := fx 2 X j (3) holdsg = @A Closure of A := A [ @A Limit points of A := fx 2 X j (1) or (3) holdg Dense: If A¯ = X then A is dense in X. Separable Space: A topological space X is separable if it contains a countable dense subset. i.e. 9A ⊂ X such that A is countable and A¯ = X. Or equivalently, there exists a sequence (xn)n2N of elements of X such that every non-empty open subset of X contains at least one element of the sequence. Limit: Let x1; x2; x3;::: be a sequence of points in X. Then limn!1 xn = y () 8 open sets U such that y 2 U, 9N > 0 such that xn 2 U 8n > N. ***** Basis for a Topology: A collection B of open sets in a topological space X is called a basis for the topology iff every open set in X is a union of sets in B. Neighbourhood of x: Any set A such that there exists an open U such that x 2 U and U ⊂ A. Subspace Topology: If Y ⊂ X then open sets in Y = open sets in X \ Y ***** 3 Continuous Functions: Let f : X ! Y where X; Y are topological spaces. Then f is continuous iff 8 open sets U in Y , f −1(U) = fx 2 X j f(x) 2 Ug is open in X. Contact Point: Let A ⊂ X where X is a metric space. Let x 2 X (may or may not be in A). If every ball B(x; r) centred at x of radius r has at least one point from A for any r > 0, then this is equivalent to calling x a Contact Point. Also, just for notational purposes, B(x; r) = fy 2 X j d(x; y) < rg Remark: Any x 2 A is a contact point of A. Dense sets in metric spaces: Suppose S ⊂ T in the metric space X. Then S is dense in T if for any x 2 T and all " > 0, there exists y 2 S such that d(x; y) < ". 1 Cauchy Sequence: fxngn=1 is cauchy () 8" > 0 ,9N 2 N such that 8m; n > N d(x; y) < ". Completeness in a metric space : A metric space X is complete () every Cauchy sequence in X converges to a limit in X.(9y 2 X such that d(xn; y) ! 0 as n ! 1) 1 Cauchy sequences in a Topological Space: In a topological space X, fxngn=1 is a Cauchy sequence () for all open sets U ⊂ X, 9N 2 N such that 8m; n > N if xn 2 U then xm 2 U. Pointwise Convergence: A sequence of functions fn on [a; b] converges pointwise to a function f, if given x 2 [a; b], then 8" > 0 9N > 0 such that 8n > N,t jfn(x) − f(x)j < ". Uniform Convergence: Consider C([a; b]; sup(`1)). Convergence for the sup distance is also called uniform convergence () fn ! g uniformly in [a; b] () 8" > 0 9N 2 N such that 8n > N 8x 2 [a; b], jfn(x) − g(x)j < ".( supx2[a;b] jfn(x) − g(x)j < "). Contraction Map: Let X be a metric space. A : X ! X is a contraction mapping if 90 < α < 1 such that d(A(x);A(y)) < αd(x; y) 8x; y 2 X. Lipschitz f : [0; 1] ! R is Lipschitz if 9K > 0 such that jf(y) − f(x)j ≤ jy − xj ***** Compactness: A topological space X is called compact if every open cover of X has a S finite subcover. That is for any collection of open sets fUαgα2I such that X = α2I Uα, Sn there exists a finite set of indices α1; : : : ; αn 2 I such that X = i=1 Uαi . S Sn A subset K ⊂ X is compact if K ⊂ α2I Uα then 9α1; : : : ; αn 2 I such that K ⊂ i=1 Uαi 4 Sequentially Compact Let X be a metric space. Then A ⊂ X is sequentially compact () (xn) a sequence in A, then 9y 2 X such that xnk ! y as n ! 1 where (xnk ) is a subsequence of (xn): That is every infinite sequence has a convergent subsequence. Product Space: Let X and Y be topological spaces. Then open sets in X × Y are generated by U × V where U is open in X and V is open in Y . This is called product topology in Y . Hausdorff Space: A topological space X is Hausdorff iff 8x1 6= x2 2 X, there exists two disjoint open sets U1, U2 such that x1 2 U1 and x2 2 U2. ***** Let F = fφ(x)g be a family of continuous functions on a finite interval [a; b] ⊂ R. Uniformly Bounded: 9M > 0 such that 8φ 2 F, jφ(x)j ≤ M 8x 2 [a; b]. Uniformly Equicontinuous: 8" > 0, 9δ > 0 such that 8x1; x2 2 [a; b] with jx1 − x2j < δ, jφ(x1) − φ(x2)j < " is true 8φ 2 F. "-net : A ⊂ X is called an "-net if for each x 2 X, there exists a y 2 A such that d(x; y) < ". If M ⊂ X, an "-net for M is a subset S ⊂ M such that: [ M ⊆ B(x; ") x2S where B(x; ") is an open ball centred at x with radius ". Totally Bounded: X is totally bounded if for each " > 0, there exists a finite "-net in X. In other words, X is totally bounded if for any " > 0, there exists a finite set x1; : : : ; xn such that [ [ X ⊂ B(x1;") ::: B(xn;") where B(x; ") is the closed ball centred at x with radius ". A set S ⊂ X is totally bounded if for each " > 0 there exists a finite "-net in S. i.e., S is totally bounded if for every " > 0 , there exists a finite subset fs1; : : : ; sng of S such that [ [ S ⊂ B(s1;") ::: B(sn;") ***** 5 Connected Sets: A topological space X is connected () X 6= U [ V where U and V are open/closed, non-empty and such that U \ V = ; . Not Connected Set: X can be written as X = U [V where U and V are open, non-empty with U \ V = ;. Path Connected: A topological space X is path connected, if 8x1; x2 2 X, there exists a function f :[a; b] ! X such that f(a) = x1, f(b) = x2 and f is continuous. Path Component: P (x) = fy 2 X j y can be connected to x by a continuous pathg ***** Rectangle: A (closed) rectangle R in Rd is given by the product of d one-dimensional closed and bounded intervals R = [a1; b1] × · · · × [ad; bd] The volume of rectangle R is denoted by jRj and given by jRj := (b1 − a1)(b2 − a2) ::: (bd − ad) Cube A cube is a rectangle with sides of equal length. So if Q ⊂ Rd is a cube of common side length `, then jQj = `d. Almost Disjoint: A union of rectangles is said to be almost disjoint if the interiors of the rectangles are disjoint.