Additional Exercises for Part I
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Niveau: Supérieur, Master
Chapter 8 Additional Exercises for Part I Analyse Master 1 : Cours de Francis Clarke (2011) 8.1 Exercise. Give an example of a lower semicontinuous function defined on a Hilbert space which is not bounded below on the unit ball. 8.2 Exercise. Let A be a bounded subset of a normed space X . Prove that co ∂A ? cl A. 8.3 Exercise. Let X be an infinite dimensional Banach space. Prove that any vector space basis for X is not countable. By considering ∞c , observe that this fact fails for infinite dimensional normed spaces that are not complete. 8.4 Exercise. Let ?n be a sequence of real numbers, and let 1 p ∞. Suppose that, for every x = (x1,x2, . . .) in l p, we have ∑n1 |?n| |xn| < ∞. Prove that the sequence ? belongs to l q, where q is the exponent conjugate to p. 8.5 Exercise. We give a direct definition of the normal cone when A is a subset of Rn, one that does not explicitly invoke polarity to the tangent cone. Let ? ? A. Show that ? ? NA(?) if and only if, for every ? > 0, there is a neighborhood V of ? such that ? ,u?? ? u?? ?u ? A ? V.
Voir
Chapter 8 Additional Exercises for Part I Analyse Master 1 : Cours de Francis Clarke (2011) 8.1 Exercise. Give an example of a lower semicontinuous function defined on a Hilbert space which is not bounded below on the unit ball. 8.2 Exercise. Let A be a bounded subset of a normed space X . Prove that co ∂A ? cl A. 8.3 Exercise. Let X be an infinite dimensional Banach space. Prove that any vector space basis for X is not countable. By considering ∞c , observe that this fact fails for infinite dimensional normed spaces that are not complete. 8.4 Exercise. Let ?n be a sequence of real numbers, and let 1 p ∞. Suppose that, for every x = (x1,x2, . . .) in l p, we have ∑n1 |?n| |xn| < ∞. Prove that the sequence ? belongs to l q, where q is the exponent conjugate to p. 8.5 Exercise. We give a direct definition of the normal cone when A is a subset of Rn, one that does not explicitly invoke polarity to the tangent cone. Let ? ? A. Show that ? ? NA(?) if and only if, for every ? > 0, there is a neighborhood V of ? such that ? ,u?? ? u?? ?u ? A ? V.
- convex
- space basis
- let
- tangent cone
- tbx ? ?
- uniform approxi- mation
- banach space
- hessian matrix ?2
Chapter 8 Additional Exercises for Part I
Analyse Master 1 : Cours de Francis Clarke (2011)
8.1 Exercise. Give an example of a lower semicontinuous function defined on a Hilbert space which is not bounded below on the unit ball. 8.2 Exercise. Let A be a bounded subset of a normed space X . Prove that co ∂ A ⊃ cl A . 8.3 Exercise. Let X be an infinite dimensional Banach space. Prove that any vector space basis for X is not countable. By considering c ∞ , observe that this fact fails for infinite dimensional normed spaces that are not complete. 8.4 Exercise. Let α n be a sequence of real numbers, and let 1 p ∞ . Suppose that, for every x = ( x 1 , x 2 , . . . ) in l p , we have ∑ n 1 | α n | | x n | < ∞ . Prove that the sequence α belongs to l q , where q is the exponent conjugate to p . 8.5 Exercise. We give a direct definition of the normal cone when A is a subset of R n , one that does not explicitly invoke polarity to the tangent cone. Let α ∈ A . Show that ζ ∈ N A ( α ) if and only if, for every ε > 0, there is a neighborhood V of α such that ζ , u − α ε u − α ∀ u ∈ A ∩ V . 8.6 Exercise. Let X be a normed space, and let C , D be closed convex subsets of X . Show by a counterexample in R 2 that C + D may not be closed. Prove that C + D is closed if one of C or D is compact. 8.7 Exercise. X is a normed space.
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140 Cours de Francis Clarke : Additional Exercises for Part I a) Let Σ and ∆ be bounded, convex, weak ∗ closed subsets of X ∗ . Prove that Σ + ∆ has the same properties. b) Let Σ i be bounded, convex, weak ∗ closed subsets of X ∗ , i = 1 , 2 , . . . , n . Prove that the set co in = 1 Σ i is weak ∗ closed. 8.8 Exercise. Let f : X × Y → R ∞ be a convex function, where X , Y are vector spaces. If, for every x ∈ X , we have g ( x ) : = inf y ∈ Y f ( x , y ) > − ∞ , then prove that g is convex. 8.9 Exercise. Let ζ : X → R be a nonzero linear functional on a normed space X . Prove that the following are equivalent: a) ζ is continuous. b) The null space N ( ζ ) : = x ∈ X : ζ , x = 0 is closed. c) N ( ζ ) is not dense in X . Deduce from this that if ζ is a linear functional on X which is not continuous at 0, then its null space is dense. 8.10 Exercise. (von Neumann) Let X = p , 1 < p < ∞ , and denote by e n (as usual) the element of X whose n -th term is 1 and whose other terms are all 0. Then, as we know, the sequence e n converges weakly, but not strongly, to 0 (Example 3.5). a) Set y n , m = e n + ne m . Prove that A : = y n , m : m > n 1 is strongly closed in X . b) Show that any weak neighborhhood of 0 contains infinitely many elements of A , but no sequence in A converges weakly to 0. Deduce that the set of all weak limits of sequences in A fails to be weakly closed; the closure of A is not obtained via convergent sequences. (This phenomenon cannot occur when a topology is metrizable.) 8.11 Exercise. Let x n be a sequence in p ( 1 < p < ∞ ) , where x n = ( x n 1 , x n 2 , . . . ) . Prove that x n converges weakly to 0 in p if and only if the sequence x n is bounded in p and, for each i , we have lim n → ∞ x ni = 0. 8.12 Exercise. Prove that in 1 , a sequence converges weakly if and only if it con-verges strongly. 8.13 Exercise. Let X = ∞ , and let C consist of those points ( x 1 , x 2 , . . . ) in X for which x i ∈ [ 0 , 1 ] ∀ i , lim i → ∞ x i = 1. Prove that C is a convex, weakly closed, bounded subset of X , but that C is not weak ∗ closed. 8.14 Exercise. Let X be a normed space whose dual ball is strictly convex: ζ 1 , ζ 2 ∈ B ∗ , ζ 1 = ζ 2 = ⇒ ζ 1 + ζ 2 / 2 ∗ < 1 .
141 Prove that the function x → x is Gaˆteaux differentiable at every nonzero point. 8.15 Exercise. Let f : X → R ∞ be convex and lsc, where X is a normed space, and suppose that lim u → ∞ f ( u ) = ∞ . Prove the existence of α > 0 and β such that f ( u ) α u − β ∀ u ∈ X . 8.16 Exercise. Let X be a normed space. a) Let C be a closed subset of X such that x , y ∈ C = ⇒ ( x + y ) / 2 ∈ C . Prove that C is convex. b) let f : X → R ∞ be a lower semicontinuous function such that f ( x + y ) / 2 12 f ( x ) + 21 f ( y ) ∀ x , y ∈ X . Prove that f is convex. 8.17 Exercise. Let f : R n → R be convex and differentiable, and suppose that, for certain positive constants a and b , we have 0 f ( x ) a + b | x | 2 ∀ x ∈ R n . Identify constants c and d such that ∇ f ( x ) c + d | x | ∀ x ∈ R n . 8.18 Exercise. Let g i : R n → R be convex and differentiable ( i = 1 , 2 , . . . , m ). We set f ( x ) = max g i ( x ) and I ( x ) = the indices i ∈ { 1 , 2 , . . . , m } : g i ( x ) = f ( x ) . 1 i m Prove that ∂ f ( x ) = co g i ( x ) : i ∈ I ( x ) . 8.19 Exercise. Let C and D be a closed convex subsets of a normed space X such that int ( C − D ) = 0/ . We prove that, for any point x in C ∩ D , we have N C ∩ D ( x ) = N C ( x ) + N D ( x ) . We may reduce to the case x = 0. Prove the inclusion ⊃ . Now, let ζ ∈ N C ∩ D ( 0 ) . Show that we can separate ( 0 , 0 ) from the set int c − d , δ − ζ , d : δ 0 , c ∈ C , d ∈ D , and conclude. 8.20 Exercise. Let f : X → R be convex, where X is a normed space. Show that the multifunction ∂ f ( ∙ ) is monotone :
142 Cours de Francis Clarke : Additional Exercises for Part I ζ 1 − ζ 2 , x 1 − x 2 0 ∀ x 1 , x 2 ∈ X , ζ i ∈ ∂ f ( x i ) ( i = 1 , 2 ) . Prove that if f : X → R isaGˆateauxdifferentiablefunctionsatisfying f G ( x 1 ) − f G ( x 2 ) , x 1 − x 2 0 ∀ x 1 , x 2 ∈ X , then f is convex. 8.21 Exercise. Let U be an open convex subset of R n , and let f : U → R be C 2 . Suppose that for all x in U with the exception of at most countably many, the Hessian matrix ∇ 2 f ( x ) has strictly positive eigenvalues. Prove that f is strictly convex. 8.22 Exercise. We demonstrate that differentiability and strict convexity are dual properties relative to conjugacy. a) Let f : R n → R be strictly convex, as well as coercive: f ( x ) / | x | → ∞ as | x | → ∞ . Prove that f ∗ is continuously differentiable. b) Let g : R n → R be convex, coercive, and differentiable. Prove that g ∗ is strictly convex. 8.23 Exercise. Let X be a Banach space that admits a bump function belonging to C b 1 , 1 ( X ) , and let f : X → R be a twice continuously differentiable function that is bounded below. Prove that for every ε > 0, there exists x ∈ X such that f ( x ) ∗ ε , f ( x ) − ε . (The latter condition means that f ( x ) v , v − ε v 2 ∀ v ∈ X . ) 8.24 Exercise. (Decrease Principle) Let f : X → R be a differentiable function on a Banach space. If f ( x ¯ ) = 0, then clearly we have, for any r > 0, inf f < f ( x ) . ¯ B ( x ¯ , r ) (This is simply Fermat’s Rule in contrapositive form.) It is possible to give a cali-brated version of this fact, as follows: Theorem. Suppose that for positive numbers δ and r we have f ( x ) ∗ δ ∀ x ∈ B ( x ¯ , r ) . Then inf f f ( x ¯ ) − δ r . B ( x ¯ , r ) Establish the following steps in the proof, which reasons from the absurd. a) If the conclusion fails, show that there exists η > 0 such that δ r − η > 0 and f ( x ¯ ) inf E f + δ r − η , where E : = B ( x ¯ , r ) .
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b) Deduce that for any λ ∈ ( 0 , r ) , there exists a point v minimizing the function w → f ( w ) + δ r λ − η w − v over E , and such that x ¯ − v < r . c) Deduce that f ( v ) ∗ ( δ r − η ) / λ , and show that this leads to a contradiction if λ has been taken to be sufficiently close to r . 8.25 Exercise. (Caristi’s fixed point theorem) Let f : E → [ 0 , ∞ ] be proper and lsc, where E , d is a complete metric space. Let g : E → E be a function such that f g ( x ) + d x , g ( x ) f ( x ) ∀ x ∈ E . Prove that g admits a fixed point. 8.26 Exercise. Let T : X → Y be a continuous linear operator, where X and Y are Banach spaces. Recall that the adjoint T ∗ of T is defined as the unique element of L ( Y ∗ , X ∗ ) satisfying T ∗ y ∗ , x = y ∗ , T x ∀ x ∈ X , y ∗ ∈ Y ∗ . The goal is to prove the following characterization of surjectivity. Theorem. The following are equivalent : 1) T is surjective. 2) For some δ > 0 , we have T B X ⊃ δ B Y . 3) For some δ > 0 , we have T ∗ y ∗ X ∗ δ y ∗ Y ∗ ∀ y ∗ ∈ Y ∗ . The proof is achieved by the following steps. a) Prove that (1) and (2) are equivalent. b) Prove that (2) implies (3). c) We assume now that (3) holds, and we prove that T is surjective. We reason by the absurd. Let y ∈ Y be a point which is not in the range T ( X ) of T . Show that, for any ε > 0, there exists x ε ∈ X which minimizes over X the function h ( x ) = T x − y Y + ε x − x ε X . d) Apply subdifferential calculus to deduce that T x ε = y if ε has been chosen suf-ficiently small. This contradiction completes the proof. 8.27 Exercise. The goal of the exercise is to prove the following uniform approxi-mation result: Theorem. Let ( K , d ) be a compact metric space, and let G be a subset of C ( K ) with the following properties :
144 Cours de Francis Clarke : Additional Exercises for Part I 1 . G is closed under addition, positive scalar multiplication, and maximum; that is, f , g ∈ G , t > 0 = ⇒ f + g , t f , and max ( f , g ) ∈ G . 2 . G contains the constant functions. 3 . For any x ∈ K and r > 0 , there exists g ∈ G such that 0 g 1 , g ( x ) = 1 , and d ( u , x ) r = ⇒ g ( u ) = 0 . Then G is dense in C ( K ) . Proof. a) Prove that the closure of G in C ( K ) inherits properties (1) to (3) above. In view of this, it suffices to prove that a closed subset G of C ( K ) having these properties must coincide with C ( K ) . Show that this follows from the following assertion: Let G be a closed subset of C ( K ) with properties (1) to (3) . Then, for any f ∈ C ( K ) and ε > 0 , there exists g ∈ G such that f ( u ) − ε g ( u ) f ( u ) ∀ u ∈ K . We proceed to prove this, arguing by contradiction. For f ∈ C ( K ) , let G f be given by G f = g ∈ G : g f . b) Prove that G f is nonempty and closed. If the assertion above fails, then for a certain f , we have inf g ∈ G f f − g = δ > 0. We proceed to show that this leads to a contradiction. c) Fix a positive ε such that ε < min ( δ / 8 , 1 / 4 ) . Prove the existence of g ∗ ∈ G f such that f − g ∗ δ + ε and f − g + ε g − g ∗ f − g ∗ δ ∀ g ∈ G f . ( ∗ ) d) Let M = { u ∈ K : f ( u ) − g ∗ ( u ) 3 δ / 4 } . Prove that M is compact, and that for each x ∈ M , there exists h x ∈ G and an open neighborhood V ( x ) of x such that 0 h x δ / 2 , h x ( x ) = δ / 2 , h x ( u ) δ / 4 ∀ u ∈ V ( x ) , and u ∈ K , ( f − g ∗ )( u ) δ / 2 = ⇒ h x ( u ) = 0 . e) Let V ( x i ) be a finite subcover of M , and set h = max h x i . Prove that 0 f ( u ) − g ∗ ( u ) − h ( u ) 7 δ / 8 ∀ u ∈ K , by considering the three cases: (i) u ∈ M , (ii) δ / 2 < ( f − g ∗ )( u ) < 3 δ / 4 , (iii) ( f − g ∗ )( u ) δ / 2 . Deduce from this that g ∗ + h ∈ G f and f − g ∗ − h 7 δ / 8.
145 f) Derive a contradiction from ( ∗ ) by taking g = g ∗ + h . 8.28 Exercise. A function f : R n → R is said to be locally convex at x if there is a neighborhood of x on which f coincides with a convex function. Let B and B ◦ be the closed and open unit balls in R n , and let G consist of those f ∈ C ( B ) which are locally convex at almost all points of B ◦ Prove that G is dense in C ( B ) . . 8.29 Exercise. The goal here is to prove the Stone-Weierstrass theorem : Theorem. Let K be a compact metric space, and let L be a linear subspace of C ( K ) with the following properties : 1 . L is closed under products ; that is : f , g ∈ L = ⇒ f g ∈ L . 2 . L contains the constant functions. 3 . L separates points : for any pair x , y of distinct points in K , there exists g ∈ L such that g ( x ) = g ( y ) . Then L is dense in C ( K ) . Proof. The proof consists of showing that the closure G of L satisfies the three properties of the theorem of Exer. 8.27, whence G = C ( K ) . Note that G is a linear subspace of C ( K ) . We require: Lemma. Given ε > 0 , there is a polynomial P in one variable such that P ( s ) − | s | < ε ∀ s ∈ [ − 1 , 1 ] . To see this, 1 let ∑ n ∞ = 0 c n t n be the binomial series for ( 1 − t ) 1 / 2 , which converges uniformly on [ 0 , 1 ] . Then we can choose N so that ( 1 − t ) 1 / 2 − Q N ( t ) < ε ∀ t ∈ [ 0 , 1 ] , where Q N ( t ) = ∑ nN = 0 c n t n . Let P ( s ) = Q N ( 1 − s 2 ) . Then P is a polynomial in s , and satisfies the required con-dition. a) Deduce from the lemma that if f ∈ L , then | f | ∈ G . b) Deduce that f , g ∈ G = ⇒ max f , g ∈ G and min f , g ∈ G . c) It remains to verify property (3) of Exer. 8.27. As a first step in this direction, prove that, for any two different points x , y ∈ K , there exists f ∈ L such that f ( x ) = 1 and f ( y ) < 0. d) Let F be a closed subset of K not containing the point x . Prove the existence of a function g ∈ G such that 0 g 1, g ( x ) = 1, and g = 0 on F . e) Deduce property (3) of Exer. 8.27, which concludes the proof. 1 The lemma is from Royden, p.173.
146 Cours de Francis Clarke : Additional Exercises for Part I 8.30 Exercise. Deduce from the Stone-Weierstrass theorem (see Exer. 8.29) that if K is a compact set in R n , then the polynomials in n variables are dense in C ( K ) . 8.31 Exercise. (Motzkin’s theorem) Let S be a nonempty closed subset of R n , and, for any x ∈ S , let proj S ( x ) denote the (nonempty) set of points u ∈ S satisfying d S ( x ) = u − x . We prove the following theorem due to Motzkin: S is convex if and only if proj S ( x ) is a singleton for each x . The necessity is known to us (see Prop. 7.3), so we turn to the hard part: showing that the uniqueness of closest points implies the convexity of S . We denote by s x the unique projection of x onto S . a) For fixed x and v , let p t be the projection of x + tv onto S . Prove that lim t ↓ 0 p t = s x . b) Show that for t > 0 we have x + tv − p t 2 − x − p t 2 d S 2 ( x + tv ) − d S 2 ( x ) x + tv − s x 2 − x − s x 2 . Deduce that the function d S 2 ( x ) isGˆateauxdifferentiableat x , with derivative 2 ( x − s x ) . c) Prove that the function ϕ ( x ) = x 2 − d S 2 ( x ) / 2 is convex. d) Let f ( x ) = x 2 / 2 + I S ( x ) . Prove that f ∗ ϕ . = e) We set g = f ∗∗ = ϕ ∗ . We prove that dom g ⊃ co S . To this end, note that g ( x ) = ϕ ∗ ( x ) = sup x ∙ y − y 2 / 2 + d S ( y ) 2 / 2 y ∈ R n 2 = y s ∈ u R p n inf S x ∙ y − y / 2 + y − s 2 / 2 s ∈ inf sup x ∙ y − y 2 / 2 + y − s 2 / 2 s ∈ S y ∈ R n = inf sup ( x − s ) ∙ y + s 2 / 2 , s ∈ S y ∈ R n which equals + ∞ if x / ∈ S , and x 2 / 2 otherwise. We deduce dom g ⊃ S , whence dom g ⊃ co S . f) Let x be a point for which ∂ g ( x ) = /0.Showthat x ∈ S . [Hint: subdifferential inversion.] g) Let A be the set of points x such that ∂ g ( x ) = 0/ . Show that S ⊃ A ⊃ dom g ⊃ co S ⊃ S . This implies S = co S , which reveals that S is convex.
147 8.32 Exercise. Let f : R n → R be convex. Prove that ∂ f is a measurable multifunc-tion. 8.33 Exercise. Let f : R n → R be continuous and have superlinear growth: u l | im f ( u ) / | u | = ∞ . | → ∞ For x ∈ R n , let Γ ( x ) be the set of points in R n which minimize the function u → f ( u ) − u • x . Prove that Γ admits a measurable selection. 8.34 Exercise. Let S be a nonempty closed subset of a Hilbert space X , with S = X . Prove the existence of a point x ∈ X at which the distance function d S fails to be differentiable. 8.35 Exercise. Let Λ : R m × R n → R be a locally Lipschitz function (the La-grangian ). We suppose that Λ is coercive in v as follows: for each x , we have L ( x , v ) / v → ∞ as v → ∞ , uniformly for x in bounded sets. We define the Hamil-tonian H as follows: H ( x , p ) = v m ∈ a R x n p , v − Λ ( x , v ) . Thus, for each x , H ( x , ∙ ) is the Fenchel conjugate of Λ ( x , ∙ ) . a) Prove that H is locally Lipschitz. b) If Λ is twice continuously differentiable and L v v ( x , v ) (the Hessian matrix with respect to v ) is positive definite at all points ( x , v ) , prove that H is continuously differentiable. c) If Λ is (jointly) convex in ( x , v ) , prove that H is concave in x (as well as convex in p ), and that ( q , p ) ∈ ∂ Λ ( x , v ) ⇐⇒ q ∈ ∂ x − H ( x , p ) , v ∈ ∂ p H ( x , p ) . 8.36 Exercise. Let E , d be a complete metric space. Given two points u , v in E , the open interval ( u , v ) refers to the set (possibly empty) of points x different from u or v such that d ( u , v ) = d ( u , x ) + d ( x , v ) . Let g : E → E be a continuous mapping satisfying, for a certain c ∈ [ 0 , 1 ) : v ∈ E , v = g ( v ) = ⇒ ∃ w ∈ v , g ( v ) such that d g ( v ) , g ( w ) c d ( v , w ) . Then g admits a fixed point. 8.37 Exercise. Let K be a compact subset of R n containing at least two points. Show that C ( K ) is not uniformly convex. 8.38 Exercise. Prove that c 0 ∗ is isometric to 1 . It follows that c 0 is not reflexive. Find an element in the set c 0 ∗∗ \ J c 0 .
