Functional analysis2

Chapter 2 Functional analysis Xiao-Min Huang mahuangxm@gdut.edu.cn Nov 2, 2020 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter 2 Outline 1 Chapter 2 What we will learn this chapter Distance Function Underlying Topology in a Metric Space and Completeness Convergence in a Metric Space and Completeness Normed Vector Spaces Underlying Metric and Topology in a Normed Space An Example of a Normed Space: Lp Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter 2 What we will learn this chapter Distance Function Underlying Topology in a Metric Space and Completeness Convergence in a Metric Space and Completeness Normed Vector Spaces Underlying Metric and Topology in a Normed Space An Example of a Normed Space: Lp Density What we will learn this chapter What is a distance (距离)? What is a metric space (度量空间)? What is a converging sequence in a metric space? What is a Cauchy sequence? What is a normed space(赋范空间)? What is a converging sequence in a normed space? What are equivalent norms? An example of Normed space: Lp Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter 2 What we will learn this chapter Distance Function Underlying Topology in a Metric Space and Completeness Convergence in a Metric Space and Completeness Normed Vector Spaces Underlying Metric and Topology in a Normed Space An Example of a Normed Space: Lp Density What we will learn this chapter What is a distance (距离)? What is a metric space (度量空间)? What is a converging sequence in a metric space? What is a Cauchy sequence? What is a normed space(赋范空间)? What is a converging sequence in a normed space? What are equivalent norms? An example of Normed space: Lp Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter 2 What we will learn this chapter Distance Function Underlying Topology in a Metric Space and Completeness Convergence in a Metric Space and Completeness Normed Vector Spaces Underlying Metric and Topology in a Normed Space An Example of a Normed Space: Lp Density What we will learn this chapter What is a distance (距离)? What is a metric space (度量空间)? What is a converging sequence in a metric space? What is a Cauchy sequence? What is a normed space(赋范空间)? What is a converging sequence in a normed space? What are equivalent norms? An example of Normed space: Lp Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter 2 What we will learn this chapter Distance Function Underlying Topology in a Metric Space and Completeness Convergence in a Metric Space and Completeness Normed Vector Spaces Underlying Metric and Topology in a Normed Space An Example of a Normed Space: Lp Density What we will learn this chapter What is a distance (距离)? What is a metric space (度量空间)? What is a converging sequence in a metric space? What is a Cauchy sequence? What is a normed space(赋范空间)? What is a converging sequence in a normed space? What are equivalent norms? An example of Normed space: Lp Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter 2 What we will learn this chapter Distance Function Underlying Topology in a Metric Space and Completeness Convergence in a Metric Space and Completeness Normed Vector Spaces Underlying Metric and Topology in a Normed Space An Example of a Normed Space: Lp Density What we will learn this chapter What is a distance (距离)? What is a metric space (度量空间)? What is a converging sequence in a metric space? What is a Cauchy sequence? What is a normed space(赋范空间)? What is a converging sequence in a normed space? What are equivalent norms? An example of Normed space: Lp Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter 2 What we will learn this chapter Distance Function Underlying Topology in a Metric Space and Completeness Convergence in a Metric Space and Completeness Normed Vector Spaces Underlying Metric and Topology in a Normed Space An Example of a Normed Space: Lp Density What we will learn this chapter What is a distance (距离)? What is a metric space (度量空间)? What is a converging sequence in a metric space? What is a Cauchy sequence? What is a normed space(赋范空间)? What is a converging sequence in a normed space? What are equivalent norms? An example of Normed space: Lp Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter 2 What we will learn this chapter Distance Function Underlying Topology in a Metric Space and Completeness Convergence in a Metric Space and Completeness Normed Vector Spaces Underlying Metric and Topology in a Normed Space An Example of a Normed Space: Lp Density What we will learn this chapter What is a distance (距离)? What is a metric space (度量空间)? What is a converging sequence in a metric space? What is a Cauchy sequence? What is a normed space(赋范空间)? What is a converging sequence in a normed space? What are equivalent norms? An example of Normed space: Lp Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter 2 What we will learn this chapter Distance Function Underlying Topology in a Metric Space and Completeness Convergence in a Metric Space and Completeness Normed Vector Spaces Underlying Metric and Topology in a Normed Space An Example of a Normed Space: Lp Density What we will learn this chapter What is a distance (距离)? What is a metric space (度量空间)? What is a converging sequence in a metric space? What is a Cauchy sequence? What is a normed space(赋范空间)? What is a converging sequence in a normed space? What are equivalent norms? An example of Normed space: Lp Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter 2 What we will learn this chapter Distance Function Underlying Topology in a Metric Space and Completeness Convergence in a Metric Space and Completeness Normed Vector Spaces Underlying Metric and Topology in a Normed Space An Example of a Normed Space: Lp Density What we will learn this chapter What is a distance (距离)? What is a metric space (度量空间)? What is a converging sequence in a metric space? What is a Cauchy sequence? What is a normed space(赋范空间)? What is a converging sequence in a normed space? What are equivalent norms? An example of Normed space: Lp Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter 2 What we will learn this chapter Distance Function Underlying Topology in a Metric Space and Completeness Convergence in a Metric Space and Completeness Normed Vector Spaces Underlying Metric and Topology in a Normed Space An Example of a Normed Space: Lp Density What we will learn this chapter What is a distance (距离)? What is a metric space (度量空间)? What is a converging sequence in a metric space? What is a Cauchy sequence? What is a normed space(赋范空间)? What is a converging sequence in a normed space? What are equivalent norms? An example of Normed space: Lp Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter 2 What we will learn this chapter Distance Function Underlying Topology in a Metric Space and Completeness Convergence in a Metric Space and Completeness Normed Vector Spaces Underlying Metric and Topology in a Normed Space An Example of a Normed Space: Lp Density Definition: Distance Function Let E be a set and d : E × E → R be a function. d is a distance function on E if i. ∀(x , y ) ∈ E × E , d(x , y ) ≥ 0; ii. ∀(x , y ) ∈ E × E , d(x , y ) = 0 ⇔ x = y ; iii. ∀(x , y ) ∈ E × E , d(x , y ) = d(y , x ); iv. ∀(x , y , z) ∈ E × E × E , d(x , y ) ≤ d(x , z) + d(z, y ). (E , d) is a metric space. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter 2 What we will learn this chapter Distance Function Underlying Topology in a Metric Space and Completeness Convergence in a Metric Space and Completeness Normed Vector Spaces Underlying Metric and Topology in a Normed Space An Example of a Normed Space: Lp Density Definition: Pseudodistance(伪距离) Function Let E be a set and d : E × E → R be a function. d is a pseudodistance function on E if i. ∀(x , y ) ∈ E × E , d(x , y ) ≥ 0; ii. ∀(x , y ) ∈ E × E , d(x , y ) = 0 ⇔ x = y ; × iii. ∀(x , y ) ∈ E × E , d(x , y ) = d(y , x ); iv. ∀(x , y , z) ∈ E × E × E , d(x , y ) ≤ d(x , z) + d(z, y ). (E , d) is a pseudometic space. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter 2 What we will learn this chapter Distance Function Underlying Topology in a Metric Space and Completeness Convergence in a Metric Space and Completeness Normed Vector Spaces Underlying Metric and Topology in a Normed Space An Example of a Normed Space: Lp Density Definition: Quadistance Function Let E be a set and d : E × E → R be a function. d is a quadistance function on E if i. ∀(x , y ) ∈ E × E , d(x , y ) ≥ 0; ii. ∀(x , y ) ∈ E × E , d(x , y ) = 0 ⇔ x = y ; iii. ∀(x , y ) ∈ E × E , d(x , y ) = d(y , x ); × iv. ∀(x , y , z) ∈ E × E × E , d(x , y ) ≤ d(x , z) + d(z, y ). (E , d) is a quasimetric space. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter 2 What we will learn this chapter Distance Function Underlying Topology in a Metric Space and Completeness Convergence in a Metric Space and Completeness Normed Vector Spaces Underlying Metric and Topology in a Normed Space An Example of a Normed Space: Lp Density Examples Consider E = Rn with n ∈ N∗ and p ∈ [1, ∞) √ dp (X , Y ) = p ∑ni=1 |Xi − Yi |p d∞ (X , Y ) = max |Xi − Yi | i∈[1,n]∩N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter 2 What we will learn this chapter Distance Function Underlying Topology in a Metric Space and Completeness Convergence in a Metric Space and Completeness Normed Vector Spaces Underlying Metric and Topology in a Normed Space An Example of a Normed Space: Lp Density Example Consider E = Rn and √ d2 (X , Y ) = ∑ni=1 |Xi − Yi |2 d2 is a distance: the Euclidean Distance Function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter 2 What we will learn this chapter Distance Function Underlying Topology in a Metric Space and Completeness Convergence in a Metric Space and Completeness Normed Vector Spaces Underlying Metric and Topology in a Normed Space An Example of a Normed Space: Lp Density Quiz Consider E ={functions from R → R defined in 0}. For f and g in E , define d(f , g) = |g(0) − f (0)| Is d a distance function on E ? No, it is a pseudodistance only: d(x 2 , x 3 ) = 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter 2 What we will learn this chapter Distance Function Underlying Topology in a Metric Space and Completeness Convergence in a Metric Space and Completeness Normed Vector Spaces Underlying Metric and Topology in a Normed Space An Example of a Normed Space: Lp Density Quiz Consider E ={functions from R → R defined in 0}. For f and g in E , define d(f , g) = |g(0) − f (0)| Is d a distance function on E ? No, it is a pseudodistance only: d(x 2 , x 3 ) = 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter 2 What we will learn this chapter Distance Function Underlying Topology in a Metric Space and Completeness Convergence in a Metric Space and Completeness Normed Vector Spaces Underlying Metric and Topology in a Normed Space An Example of a Normed Space: Lp Density Quiz Consider a set E . For all x and y in E , define d(x , y ) = 0 if x = y ; d(x , y ) = 1 if x ̸= y ; Is d a distance function on E ? Yse. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter 2 What we will learn this chapter Distance Function Underlying Topology in a Metric Space and Completeness Convergence in a Metric Space and Completeness Normed Vector Spaces Underlying Metric and Topology in a Normed Space An Example of a Normed Space: Lp Density Quiz Consider a set E . For all x and y in E , define d(x , y ) = 0 if x = y ; d(x , y ) = 1 if x ̸= y ; Is d a distance function on E ? Yse. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter 2 What we will learn this chapter Distance Function Underlying Topology in a Metric Space and Completeness Convergence in a Metric Space and Completeness Normed Vector Spaces Underlying Metric and Topology in a Normed Space An Example of a Normed Space: Lp Density Distance from a Point to a Set Let E be a metric space with distance d. Let a ∈ E . Let X ⊆ E . The distance between a and X is defined by d(a, X ) = inf {d(a, X ), x ∈ X }. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter 2 What we will learn this chapter Distance Function Underlying Topology in a Metric Space and Completeness Convergence in a Metric Space and Completeness Normed Vector Spaces Underlying Metric and Topology in a Normed Space An Example of a Normed Space: Lp Density From a Metric to a Topological Space Let E be a metric space. Given x ∈ E . define the open ball around x with radius r > 0 by: Br (x ) = {y ∈ E | d(x , y ) < r } Define a topology on E by T = {O ⊆ E |∀x ∈ O, ∃r > 0, Br (x ) ⊆ O}. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter 2 What we will learn this chapter Distance Function Underlying Topology in a Metric Space and Completeness Convergence in a Metric Space and Completeness Normed Vector Spaces Underlying Metric and Topology in a Normed Space An Example of a Normed Space: Lp Density Quiz Prove T is a topology. T = {O ⊆ E |∀x ∈ O, ∃r > 0, Br (x ) ⊆ O} i. 0, / E ∈ T. ii. Any union of elements of T is in T . iii. Any finite intersection of elements of T is in T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter 2 What we will learn this chapter Distance Function Underlying Topology in a Metric Space and Completeness Convergence in a Metric Space and Completeness Normed Vector Spaces Underlying Metric and Topology in a Normed Space An Example of a Normed Space: Lp Density From a Metric to a Topological Space Given a metric space, we can derive an associated topological space. It is a Normal Hausdorff Space. Given a topological space, we may not always find a distance from which the topology derive. When it is possible, the space is metrizable. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter 2 What we will learn this chapter Distance Function Underlying Topology in a Metric Space and Completeness Convergence in a Metric Space and Completeness Normed Vector Spaces Underlying Metric and Topology in a Normed Space An Example of a Normed Space: Lp Density Quiz Find a topology which is not metrizable. Trivial topology. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter 2 What we will learn this chapter Distance Function Underlying Topology in a Metric Space and Completeness Convergence in a Metric Space and Completeness Normed Vector Spaces Underlying Metric and Topology in a Normed Space An Example of a Normed Space: Lp Density Quiz Find a topology which is not metrizable. Trivial topology. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter 2 What we will learn this chapter Distance Function Underlying Topology in a Metric Space and Completeness Convergence in a Metric Space and Completeness Normed Vector Spaces Underlying Metric and Topology in a Normed Space An Example of a Normed Space: Lp Density Converging Sequences Let (X , d) be a metric space. (now also a Hausdorff topological space) Let (xn ) be a sequence of elements of X . We say that (xn ) converges to l if ∀V ∈ V (l), ∃N ∈ N, n ≥ N ⇒ xn ∈ V . It is equivalent to ∀ε > 0, ∃N ∈ N, n ≥ N ⇒ d(xn , l) < ε . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter 2 What we will learn this chapter Distance Function Underlying Topology in a Metric Space and Completeness Convergence in a Metric Space and Completeness Normed Vector Spaces Underlying Metric and Topology in a Normed Space An Example of a Normed Space: Lp Density Example xn = 1/n2 Prove (xn ) converges to 0. Let ε > 0. Let N = [1/ε 1/2 ] + 1. Then n > N implies n > 1/ε 1/2 . Thus 1/n2 < ε . Thus d(xn , 0) < ε . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter 2 What we will learn this chapter Distance Function Underlying Topology in a Metric Space and Completeness Convergence in a Metric Space and Completeness Normed Vector Spaces Underlying Metric and Topology in a Normed Space An Example of a Normed Space: Lp Density Example xn = 1/n2 Prove (xn ) converges to 0. Let ε > 0. Let N = [1/ε 1/2 ] + 1. Then n > N implies n > 1/ε 1/2 . Thus 1/n2 < ε . Thus d(xn , 0) < ε . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter 2 What we will learn this chapter Distance Function Underlying Topology in a Metric Space and Completeness Convergence in a Metric Space and Completeness Normed Vector Spaces Underlying Metric and Topology in a Normed Space An Example of a Normed Space: Lp Density Example xn = 1/n2 Prove (xn ) converges to 0. Let ε > 0. Let N = [1/ε 1/2 ] + 1. Then n > N implies n > 1/ε 1/2 . Thus 1/n2 < ε . Thus d(xn , 0) < ε . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter 2 What we will learn this chapter Distance Function Underlying Topology in a Metric Space and Completeness Convergence in a Metric Space and Completeness Normed Vector Spaces Underlying Metric and Topology in a Normed Space An Example of a Normed Space: Lp Density Definition: Completeness In a metric space, we call Cauchy sequence, a sequence (un ) s.t. ∀ε > 0, ∃N > 0, m > n ≥ N ⇒ d(un , um ) < ε . A metric space X is called complete, if all Cauchy sequences of elements of X converge. R is complete. Q isn’t. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter 2 What we will learn this chapter Distance Function Underlying Topology in a Metric Space and Completeness Convergence in a Metric Space and Completeness Normed Vector Spaces Underlying Metric and Topology in a Normed Space An Example of a Normed Space: Lp Density Example xn = 1/n2 Prove (xn ) is a Cauchy sequence. Let ε > 0. Let N = [2/ε ] + 1. Then q > p > N implies q > p > 2/ε implies 1/p < ε /2. Thus |1/p − 1/q|(1/p + 1/q) ≤ 2|1/p − 1/q| < 2/p < ε . Thus |1/q 2 − 1/p 2 | < ε . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter 2 What we will learn this chapter Distance Function Underlying Topology in a Metric Space and Completeness Convergence in a Metric Space and Completeness Normed Vector Spaces Underlying Metric and Topology in a Normed Space An Example of a Normed Space: Lp Density Example xn = 1/n2 Prove (xn ) is a Cauchy sequence. Let ε > 0. Let N = [2/ε ] + 1. Then q > p > N implies q > p > 2/ε implies 1/p < ε /2. Thus |1/p − 1/q|(1/p + 1/q) ≤ 2|1/p − 1/q| < 2/p < ε . Thus |1/q 2 − 1/p 2 | < ε . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter 2 What we will learn this chapter Distance Function Underlying Topology in a Metric Space and Completeness Convergence in a Metric Space and Completeness Normed Vector Spaces Underlying Metric and Topology in a Normed Space An Example of a Normed Space: Lp Density example √ Let xn = [10n 2]/10n . x0 = 1. x1 = 1.4. x2 = 1.41. x3 = 1.414. (xn ) is a Cauchy sequence. √ Its limit is 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter 2 What we will learn this chapter Distance Function Underlying Topology in a Metric Space and Completeness Convergence in a Metric Space and Completeness Normed Vector Spaces Underlying Metric and Topology in a Normed Space An Example of a Normed Space: Lp Density Definition: Norm Let E be a vector space and N : E → R a function. N is a norm on E if i. ∀x ∈ E , N(x ) = 0 ⇔ x = 0. ii. ∀(x , λ ) ∈ E × R, N(λ x ) = |λ |N(x ). iii. ∀(x , y ) ∈ E × E , N(x + y ) ≤ N(x ) + N(y ). Assertions (ii) and (iii) imply N(x ) ≥ 0 for all x in E . (E , N) is a normed vector space. N(x ) is usually noted ||x ||E or simply ||x ||. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter 2 What we will learn this chapter Distance Function Underlying Topology in a Metric Space and Completeness Convergence in a Metric Space and Completeness Normed Vector Spaces Underlying Metric and Topology in a Normed Space An Example of a Normed Space: Lp Density Definition: Seminorm Let E be a vector space and N : E → R a function. N is a seminorm on E if i. ∀x ∈ E , N(x ) = 0 ⇔ x = 0. × ii. ∀(x , λ ) ∈ E × R, N(λ x ) = |λ |N(x ). iii. ∀(x , y ) ∈ E × E , N(x + y ) ≤ N(x ) + N(y ). Nevertheless, assertions (ii) implies N(x ) = 0. (E , N) is a seminormed vector space. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter 2 What we will learn this chapter Distance Function Underlying Topology in a Metric Space and Completeness Convergence in a Metric Space and Completeness Normed Vector Spaces Underlying Metric and Topology in a Normed Space An Example of a Normed Space: Lp Density Examples of norms Consider E = ℓ∞ ={bounded sequences}. For u in E , define N(u) = sup{|ui |, i ∈ N} N(u) iff u = 0. N(λ u) = |λ |N(u), for any real number λ . N(u + v ) ≤ N(u) + N(v ). (E , N) is a normed space. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter 2 What we will learn this chapter Distance Function Underlying Topology in a Metric Space and Completeness Convergence in a Metric Space and Completeness Normed Vector Spaces Underlying Metric and Topology in a Normed Space An Example of a Normed Space: Lp Density Examples of norms Consider E = Rn with n ∈ N∗ and p ∈ [1, ∞). √ ||X ||p = n p ∑ |Xi |p = dp (X , 0). i=1 ||X ||∞ = max = d∞ (X , 0). i∈[1,n]∩N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter 2 What we will learn this chapter Distance Function Underlying Topology in a Metric Space and Completeness Convergence in a Metric Space and Completeness Normed Vector Spaces Underlying Metric and Topology in a Normed Space An Example of a Normed Space: Lp Density Quiz Consider E = {functions from R to R defined in0}. For f in E , define N(f ) = d(f , 0) = |f (0)|. Is N a norm on E ? No. It is a seminorm only: ||x 2 || = 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter 2 What we will learn this chapter Distance Function Underlying Topology in a Metric Space and Completeness Convergence in a Metric Space and Completeness Normed Vector Spaces Underlying Metric and Topology in a Normed Space An Example of a Normed Space: Lp Density Quiz Consider E = {functions from R to R defined in0}. For f in E , define N(f ) = d(f , 0) = |f (0)|. Is N a norm on E ? No. It is a seminorm only: ||x 2 || = 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter 2 What we will learn this chapter Distance Function Underlying Topology in a Metric Space and Completeness Convergence in a Metric Space and Completeness Normed Vector Spaces Underlying Metric and Topology in a Normed Space An Example of a Normed Space: Lp Density From a Normed to a Metric Space Let (E , N) be a normed space. Given x and y in E , define d(x , y ) = N(y − x ) (E , d) is a metric space. The unit open ball associated to the norm is B = B(0, 1) = {x ∈ E | N(x ) < 1}. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter 2 What we will learn this chapter Distance Function Underlying Topology in a Metric Space and Completeness Convergence in a Metric Space and Completeness Normed Vector Spaces Underlying Metric and Topology in a Normed Space An Example of a Normed Space: Lp Density From a Normed to a Metric Space Let (E , N) be a normed space. Given x and y in E , define d(x , y ) = N(y − x ) (E , d) is a metric space. The unit open ball associated to the norm is B = B(0, 1) = {x ∈ E | N(x ) < 1}. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter 2 What we will learn this chapter Distance Function Underlying Topology in a Metric Space and Completeness Convergence in a Metric Space and Completeness Normed Vector Spaces Underlying Metric and Topology in a Normed Space An Example of a Normed Space: Lp Density From a Normed to a Metric Space Let (E , N) be a normed space. Given x and y in E , define d(x , y ) = N(y − x ) (E , d) is a metric space. The unit open ball associated to the norm is B = B(0, 1) = {x ∈ E | N(x ) < 1}. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter 2 What we will learn this chapter Distance Function Underlying Topology in a Metric Space and Completeness Convergence in a Metric Space and Completeness Normed Vector Spaces Underlying Metric and Topology in a Normed Space An Example of a Normed Space: Lp Density From a Normed to a Metric Space Let (E , N) be a normed space. Given x and y in E , define d(x , y ) = N(y − x ) (E , d) is a metric space. The unit open ball associated to the norm is B = B(0, 1) = {x ∈ E | N(x ) < 1}. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter 2 What we will learn this chapter Distance Function Underlying Topology in a Metric Space and Completeness Convergence in a Metric Space and Completeness Normed Vector Spaces Underlying Metric and Topology in a Normed Space An Example of a Normed Space: Lp Density From a Normed to a Metric Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter 2 What we will learn this chapter Distance Function Underlying Topology in a Metric Space and Completeness Convergence in a Metric Space and Completeness Normed Vector Spaces Underlying Metric and Topology in a Normed Space An Example of a Normed Space: Lp Density From a Normed to a Metric Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter 2 What we will learn this chapter Distance Function Underlying Topology in a Metric Space and Completeness Convergence in a Metric Space and Completeness Normed Vector Spaces Underlying Metric and Topology in a Normed Space An Example of a Normed Space: Lp Density From a Normed to a Metric Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter 2 What we will learn this chapter Distance Function Underlying Topology in a Metric Space and Completeness Convergence in a Metric Space and Completeness Normed Vector Spaces Underlying Metric and Topology in a Normed Space An Example of a Normed Space: Lp Density From a Normed to a Metric Space Given a normed space, we can derive an associated metric space d(x , y ) = ||x − y ||. Given a metric space, It may not be a linear space. Even if it is a linear space, There may be no norm inducing the distance. For example: d(x , x ) = 0 and d(x , y ) = 1 for x ̸= y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter 2 What we will learn this chapter Distance Function Underlying Topology in a Metric Space and Completeness Convergence in a Metric Space and Completeness Normed Vector Spaces Underlying Metric and Topology in a Normed Space An Example of a Normed Space: Lp Density From a Normed to a Metric Space Given a normed space, we can derive an associated metric space d(x , y ) = ||x − y ||. Given a metric space, It may not be a linear space. Even if it is a linear space, There may be no norm inducing the distance. For example: d(x , x ) = 0 and d(x , y ) = 1 for x ̸= y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter 2 What we will learn this chapter Distance Function Underlying Topology in a Metric Space and Completeness Convergence in a Metric Space and Completeness Normed Vector Spaces Underlying Metric and Topology in a Normed Space An Example of a Normed Space: Lp Density Topological, Metric and Normed Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter 2 What we will learn this chapter Distance Function Underlying Topology in a Metric Space and Completeness Convergence in a Metric Space and Completeness Normed Vector Spaces Underlying Metric and Topology in a Normed Space An Example of a Normed Space: Lp Density Converging Sequences Let (X , N) be a normed space. Let (xn ) be a sequence of elements of X . We say that (xn ) converges to l if ∀ε > 0, ∃N ∈ N, n ≥ N ⇒ xn ∈ Bε (l). It is equivalent to ∀ε > 0, ∃N ∈ N, n ≥ N ⇒ ||xn − l|| < ε . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter 2 What we will learn this chapter Distance Function Underlying Topology in a Metric Space and Completeness Convergence in a Metric Space and Completeness Normed Vector Spaces Underlying Metric and Topology in a Normed Space An Example of a Normed Space: Lp Density Remark Remember last chapter? The norm is a continuous function. |N(xn ) − N(l)| ≤ N(xn − l) = d(xn , l) If (xn ) converges to l then N(xn ) converges to N(l). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter 2 What we will learn this chapter Distance Function Underlying Topology in a Metric Space and Completeness Convergence in a Metric Space and Completeness Normed Vector Spaces Underlying Metric and Topology in a Normed Space An Example of a Normed Space: Lp Density Definition: Strength of a Norm Let E be a vector space. Na is stronger than Nb if there exists a non-negative constant Ca Such that for all x in E , Nb (x ) ≤ Ca Na (x ). The balls of Na can be included in the balls Nb . (after a possible homothetic transformation) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter 2 What we will learn this chapter Distance Function Underlying Topology in a Metric Space and Completeness Convergence in a Metric Space and Completeness Normed Vector Spaces Underlying Metric and Topology in a Normed Space An Example of a Normed Space: Lp Density Definition: Norm Equivalence Let E be a vector space. Na and Nb are equivalent if there exists two non-negative constant C1 and C2 Such that for all x in E , C1 Na ≤ Nb (x ) ≤ C2 Na (x ). Norms are equivalent iff: associated balls can be included in one another. (after a possible homothetic transformation) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter 2 What we will learn this chapter Distance Function Underlying Topology in a Metric Space and Completeness Convergence in a Metric Space and Completeness Normed Vector Spaces Underlying Metric and Topology in a Normed Space An Example of a Normed Space: Lp Density Norm Equivalence Theorem Let E be a finite-dimensional vector space. All norms on E are equivalent. Corollary Let E be a finite-dimensional vector space. There is only one topology induced by the norms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter 2 What we will learn this chapter Distance Function Underlying Topology in a Metric Space and Completeness Convergence in a Metric Space and Completeness Normed Vector Spaces Underlying Metric and Topology in a Normed Space An Example of a Normed Space: Lp Density The usual topology of Rn There is only one topology induced by the norms. It is called the usual topology of Rn . What do the open sets look like? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter 2 What we will learn this chapter Distance Function Underlying Topology in a Metric Space and Completeness Convergence in a Metric Space and Completeness Normed Vector Spaces Underlying Metric and Topology in a Normed Space An Example of a Normed Space: Lp Density Norm Equivalence Recall Br (x ) = {y ∈ X | ||x − y || < r } T = {O ⊆ E | ∀x ∈ O, ∃r > 0, Br (x ) ⊆ O} A stronger norm will provide a stronger topology. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter 2 What we will learn this chapter Distance Function Underlying Topology in a Metric Space and Completeness Convergence in a Metric Space and Completeness Normed Vector Spaces Underlying Metric and Topology in a Normed Space An Example of a Normed Space: Lp Density Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter 2 What we will learn this chapter Distance Function Underlying Topology in a Metric Space and Completeness Convergence in a Metric Space and Completeness Normed Vector Spaces Underlying Metric and Topology in a Normed Space An Example of a Normed Space: Lp Density Integration We will consider Ω an open set of Rn equipped with the Lebesgue measure. The set of Lebesgue-integrable functions from Ω to R will be noted L1 (Ω) or simply L1 when no confusion is possible. Functions that are equal almost everywhere ∫are "identified". We note: ||f ||L1 = Ω |f (x )|dx . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter 2 What we will learn this chapter Distance Function Underlying Topology in a Metric Space and Completeness Convergence in a Metric Space and Completeness Normed Vector Spaces Underlying Metric and Topology in a Normed Space An Example of a Normed Space: Lp Density Definition We note Lp the set of measurable functions. from Ω to R whose p-th power belongs to L1 (Ω). Functions equal almost everywhere are identified. We note Lp when no confusion is possible. We note ||f ||Lp = ||f ||p = √ ∫ p p Ω |f (x )| dx . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter 2 What we will learn this chapter Distance Function Underlying Topology in a Metric Space and Completeness Convergence in a Metric Space and Completeness Normed Vector Spaces Underlying Metric and Topology in a Normed Space An Example of a Normed Space: Lp Density Definition We note L∞ (Ω) the set of measurable functions. from Ω to R for which there exists a real number C . s.t. for almost every x in Ω, |f (x )| ≤ C . Functions equal almost everywhere are identified. We note L∞ when no confusion is possible. We note ||f ||L∞ = ||f ||∞ = inf{C , |f (x )| ≤ C a.e.on Ω}. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter 2 What we will learn this chapter Distance Function Underlying Topology in a Metric Space and Completeness Convergence in a Metric Space and Completeness Normed Vector Spaces Underlying Metric and Topology in a Normed Space An Example of a Normed Space: Lp Density Definition Let p ∈ [1, ∞]. A function f belongs to Lploc (Ω) when f 1K belongs to Lp (Ω) for every compact K ⊆ Ω. (1K is the characteristic function of K : 1K (x ) = 1 if x ∈ K and 0 otherwise) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter 2 What we will learn this chapter Distance Function Underlying Topology in a Metric Space and Completeness Convergence in a Metric Space and Completeness Normed Vector Spaces Underlying Metric and Topology in a Normed Space An Example of a Normed Space: Lp Density Definition Let p ∈ [1, ∞]. We call Hölder conjugate (or dual index) of p, the number p ′ = 1 + 1/(p − 1) so that 1/p + 1/p ′ = 1 (if p = 1 then p ′ = ∞ and p = ∞ then p ′ = 1) Note that the Hölder conjugate of 2 is 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter 2 What we will learn this chapter Distance Function Underlying Topology in a Metric Space and Completeness Convergence in a Metric Space and Completeness Normed Vector Spaces Underlying Metric and Topology in a Normed Space An Example of a Normed Space: Lp Density Norm on Lp Proposition (Hölder’s inequality) Let p ∈ [1, ∞] and p ′ be its Hölder conjugate. ′ Let f ∈ Lp and g ∈ Lp . Then fg ∈ L1 and ||fg||1 ≤ ||f ||p ||g||p ′ . Corollary Let p ∈ [1, ∞]. ′ || × ||p is a norm on Lp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter 2 What we will learn this chapter Distance Function Underlying Topology in a Metric Space and Completeness Convergence in a Metric Space and Completeness Normed Vector Spaces Underlying Metric and Topology in a Normed Space An Example of a Normed Space: Lp Density Interpolation Inequality Proposition Let {fi , i ∈ I}be a family functions with fi ∈ Lpi . and 1/p = ∑ 1/pi ≤ 1. Then ∏ fi ∈ Lp (Ω) and || ∏ fi ||p ≤ ∏ ||fi ||pi . Corollary If f ∈ Lp ∩ Lq , then f ∈ Lr for any r s.t. p ≤ r ≤ q. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter 2 What we will learn this chapter Distance Function Underlying Topology in a Metric Space and Completeness Convergence in a Metric Space and Completeness Normed Vector Spaces Underlying Metric and Topology in a Normed Space An Example of a Normed Space: Lp Density Approximation of π . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter 2 What we will learn this chapter Distance Function Underlying Topology in a Metric Space and Completeness Convergence in a Metric Space and Completeness Normed Vector Spaces Underlying Metric and Topology in a Normed Space An Example of a Normed Space: Lp Density Q is dense in R Approximation of π There exist a and b in Q with |b − a| as small as desired (but not 0) such that a < π < b. For any x in R There exist a and b in Q with |b − a| as small as desired (but not 0) such that a < x < b. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter 2 What we will learn this chapter Distance Function Underlying Topology in a Metric Space and Completeness Convergence in a Metric Space and Completeness Normed Vector Spaces Underlying Metric and Topology in a Normed Space An Example of a Normed Space: Lp Density Density Let (x , d) be a metric space. Y ⊆X Y is dense in X if for all x ∈ X and for all ε > 0, there exists y ∈ Y s.t. d(x , y ) < ε . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter 2 What we will learn this chapter Distance Function Underlying Topology in a Metric Space and Completeness Convergence in a Metric Space and Completeness Normed Vector Spaces Underlying Metric and Topology in a Normed Space An Example of a Normed Space: Lp Density Density Let (x , d) be a metric space. Y ⊆X Y is dense in X if for all x ∈ X there exists a sequence (yn ) of elements of Y converging to x . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter 2 What we will learn this chapter Distance Function Underlying Topology in a Metric Space and Completeness Convergence in a Metric Space and Completeness Normed Vector Spaces Underlying Metric and Topology in a Normed Space An Example of a Normed Space: Lp Density Thank you for your attention! . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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