Convexity and Smoothness

Transcription

1 Capter 4 Convexity and Smootness 4. Strict Convexity, Smootness, and Gateaux Di erentiablity Definition 4... Let X be a Banac space wit a norm denoted by k k. A map f : X \{0}!X \{0}, f 7! f x is called a support mapping wenever. a) f( x) = f x, for >0 and b) If x 2 S X,tenkf x k = and f x (x) = (and tus f x (x) = 2 for all x 2 X). Often we only define f x for x 2 S X and ten assume tat f x = f x/, for all x 2 X \{0}. For x 2 X a support functional of x is an element x 2 X,witkx k = and x,xi = 2. Tus a support map is a map f ( ) : X! X,wic assigns to eac x2x a support functional of x. We say tat X is smoot at x 0 2 S X if tere exists a unique f x 2 S X, for wic f x (x) =, and we say tat X is smoot if it is smoot at eac point of S X. Te Banac space X is said to ave Gateaux di erentiable norm at x 0 2 S X, if for all y 2 S X 0 (x 0,y)=lim!0 kx 0 + yk kx 0 k exists, and we say tat k kis Gateaux di erentiable if it is Gateaux di erentiable norm at eac x 0 2 S X. 09

2 0 CHAPTER 4. CONVEXITY AND SMOOTHNESS Example For X = L p [0, ], <p< te function f : L p [0, ]! L q [0, ], f x (t) = sign(x(t)) x(t) p p/q p = p p q x(t) p q is a (and te only) support function for L p [0, ]. In order establis a relation between Gateaux di erentiability and smootness we observe te following equalities and inequalities for any x 2 X, y 2 S X, and >0: = f x(y) = f x(x) 2 + = f x(x + y) 2 f x(x + y) 2 kf xkkx + yk 2 kx + yk = = kx + yk2 kx + yk kx + yk kx + yk2 f kx + yk x+y (x) = f x+y(x + y) f x+y (x) kx + yk = f x+y(y)+f x+y (x) f x+y (x) kx + yk f x+y(y) kx + yk = f x+y(y) kx + yk and tus for any x 2 X,y 2S X, and >0: (4.) f x(x + y) kx + yk f x+y(y) kx + yk.

3 4.. STRICT CONVEXITY, SMOOTHNESS, AND GATEAUX DIFFERENTIABLITY Teorem Assume X is a Banac space and x 0 2 S X. Te following statements are equivalent: a) X is smoot at x 0. b) Every support mapping f : x 7! f x is norm to w continuous from S X to S X at te point x 0. c) Tere exists a support mapping f ( ) : x 7! f x wic is norm to w continuous from S X to S X at te point x 0. d) Te norm is Gateaux di erentiable at x 0. In tat case = 0 (x 0,y)=lim!0 kx 0 + yk kx 0 k for all y 2 S X. Proof. (b) ) (a). Assume tat (x i ) i2i S X is a net, wic converges in norm to x 0, but for wic f xn does not converge in w to f x0,were f ( ) : X! X is a support map. After passing to a subnet we can assume by Alaoglu s Teorem tat (f xi ) converges in w to some x 2 B X (wic is not f x0 ). As x (x 0 ) = x (x 0 ) f xi (x i ) x (x 0 ) f xi (x 0 ) + f xi (x 0 x n ) x (x 0 ) f xi (x 0 ) + kx 0 x i k! i2i 0, it follows tat x (x 0 ) =, and since kx kwemustavekx k =. Since x 6= f x0, X cannot be smoot at x 0. (b) ) (c) is clear (since by Te Teorem of Han Banac tere is always at least one support map). (c) ) (d) Follows from (4.), and from applying (4.) to y instead of y wic gives kx 0 yk kx 0 k kx 0 k = kx 0 + ( y)k kx 0 k kx 0 k f x 0 ( y) kx 0 k = f x0 (y) and kx 0 yk kx 0 k kx 0 k = kx 0 + ( y)k kx 0 k kx 0 k f x0 +( y)( y) kx 0 + ( y)k = f x 0 +( y)(y) kx 0 + ( y)k.

4 2 CHAPTER 4. CONVEXITY AND SMOOTHNESS (d) ) (a) Let f 2 S x be suc tat f(x 0 )=kx 0 k =. Since (4.) is true for any support function it follows tat and f(y) kx 0 + yk kx 0 yk kx 0 k kx 0 k, for all y 2S X and >0, = kx 0 +( y)k kx 0 k f( y) =f(y) for all y 2 S X and >0. Tus, by assumption (d), 0 (x 0,y)=f(y), wic proves te uniqueness of f 2 S X wit f(x 0 ) =. Definition A Banac space X wit norm k k is called strictly convex wenever S(X) contains no non-trivial line segement, i.e. if for all x, y 2 S X, x 6= y it follows tat kx + yk < 2. Teorem If X is strictly convex ten X is smoot, and if X is smoot te X is strictly convex. Proof. If X is not smoot ten tere exists an x 0 2 S X, and two functionals x 6= y in S X wit x (x 0 )=y (x 0 ) = but tis means tat kx + y k (x + y )(x 0 )=2, wic implies tat X is not strictly convex. If X is not strictly convex ten tere exist x 6= y in S X so tat k x +( )yk =, for all 0. So let x 2 S X suc tat x x + y =. 2 But tis implies tat =x x + y = 2 2 x (x)+ 2 x (y) =, wic implies tat x (x) =x (y) =, wic by viewing x and y to be elements in X, implies tat X is not smoot. Exercises. Sow tat ` admits an equivalent norm wic is strictly convex and (`, ) is (isometrically) te dual of c 0 wit some equivalent norm.

5 4.. STRICT CONVEXITY, SMOOTHNESS, AND GATEAUX DIFFERENTIABLITY3 2. Assume tat T : X! Y is a linear, bounded, and injective operator between two Banac spaces and assume tat Y is strictly convex. Sow tat X admits an equivalent norm for wic X is strictly convex.

6 4 CHAPTER 4. CONVEXITY AND SMOOTHNESS 4.2 Uniform Convexity and Uniform Smootness Definition Let X be a Banac space wit norm k k. We say tat te norm of X is Frécet di erentiable at x 0 2 S X if lim!0 kx 0 + yk kx 0 k exists uniformly in y 2 S X. We say tat te norm of X is Frécet di erentiable if te norm of X is Frécet di erentiable at eac x 0 2 S X. Remark. By Teorem 4..3 it follows from te Frecét di erentiability of te norm at x 0 tat tere a unique support functional f x0 2 S X and lim!0 kx 0 + yk kx 0 k f x0 (y) =0, uniformly in y and tus tat (put z = y) kx 0 + zk kx 0 k f x0 (z) lim =0. z!0 kzk In particular, if X as a Frécet di erentiable norm it follows from Teorem 4..3 tat tere is a unique support map x! f x. Proposition Let X be a Banac space wit norm k k. Ten te norm is Frécet di erentiable if and only if te support map is norm-norm continuous. Proof. (We assume tat K = R) ) Assume tat (x n ) S X converges to x 0 and put x n = f xn, n 2 N, and x 0 = f x 0. It follows from Teorem 4..3 tat x n(x 0 )!, for n!. Assume tat our claim were not true, and we can assume tat for some ">0weavekx n x 0k > 2", and terefore we can coose vectors z n 2 S X, for eac n 2 N so tat (x n x 0 )(z n) > 2". But ten x 0(x 0 ) x n(x 0 ) x 0(x 0 ) x n(x 0 ) " x n(z n ) x 0(z n ) {z } >2" = x n(x 0 ) x 0(x 0 ) + " x 0(z n ) x n(z n ) x n(x 0 ) x 0(x 0 ) = x n x 0 x 0 + z n " x 0(x 0 ) x n(x 0 )

7 4.2. UNIFORM CONVEXITY AND UNIFORM SMOOTHNESS 5 Tus if we put x n x 0 + z n " x 0(x 0 ) x n(x 0 ) x 0 x 0 + z n " x 0(x 0 ) x n(x 0 ) x 0 + z n " x 0(x 0 ) x n(x 0 ) kx 0 k x 0 z n " x 0(x 0 ) x n(x 0 ) y n = z n " x 0(x 0 ) x n(x 0 ), it follows tat ky n k!0, if n!, and, using te Frécet di erentiability of te norm tat (note tat x 0 (x 0) x n(x 0 ) /ky n k = ") we deduce tat. 0 <"= x 0 (x 0) x n(x 0 ) ky n k kx 0 + y n k kx 0 k x 0 (y n) ky n k wic is a contradiction. ( From (4.) it follows tat for x, y 2 S X, and 2 R! n! 0, kx + yk f x+y(y) kx + yk f x+y (y) + f x+y(y) kx + yk f x+y (y) kf x+y f x k + + kf x+y k, wic converges uniformly in y to 0 and proves our claim. Definition Let X be a Banac space wit norm k k. We say tat te norm is uniformly Frécet di erentiable on S X if lim!0 kx + yk, uniformly in x 2 S X and y 2 S X. In oter words if for all ">0tereisa >0 so tat for all x, y 2 S X and all 2 R, 0< < kx + yk <".

8 6 CHAPTER 4. CONVEXITY AND SMOOTHNESS X is uniformly convex if for all ">0tereisa >0 so tat for all x, y 2 S X wit kx yk " it follows tat k(x + y)/2k <. We call ( ) X(") =inf kx + yk : x, y 2 S X, kx 2 yk ", for " 2 [0, 2] te modulus of uniform convexity of X. X is called uniform smoot if for all ">0tereexistsa for all x, y 2 S X and all 2 (0, ] >0 so tat kx + yk + kx yk < 2+". Te modulus of uniform smootness of X is te map :[0, )! [0, ) ( ) kx + zk kx zk X ( ) =sup + :x, z 2 X, =, kzk. 2 2 Remark. X is uniformly convex if and only if X(") > 0 for all ">0. X is uniformly smoot if and only if lim!0 X ( )/ = 0. Teorem For a Banac space X te following statements are equivalent. a) Tere exists a support map x! f x wic uniformly continuous on S X wit respect to te norms. b) Te norm on X is uniformly Frécet di erentiable on S X. c) X is uniformly smoot. d) X is uniformly convex. e) Every support map x! f x is uniformly continuous on S X wit respect to te norms. Proof. (a))(b) We proceed as in te proof of Proposition From (4.) it follows tat for x, y 2 S X, and 2 R kx + yk f x+y(y) kx + yk

9 4.2. UNIFORM CONVEXITY AND UNIFORM SMOOTHNESS 7 f x+y (y) + f x+y(y) kx + yk f x+y (y) kf x+y f x k + + kf x+y k wic converges by (a) uniformly in x and y, to 0. (b))(c). Assuming (b) we can coose for ">0a 2 (0, ) and all x, y 2 S X >0 so tat for all kx + yk <"/2. But tis implies tat for all 2 (0, ) and all x, y 2 S X we ave kx + yk + kx =2+ 2+", yk kx + yk kx + ( y)k +! f x ( y) wic implies our claim. (c))(d). Let ">0. By (c) we can find >0 suc tat for all x 2 S X and z 2 X, witkzk,weavekx + zk + kx zk 2+"kzk/4. Let x,y 2 S X wit kx y k ". Tere is a z 2 X, kzk /2 so tat (x y )(z) " /2. Tis implies kx + y k = sup (x + y )(x) x2s X = sup x2s X x (x + z)+y (x z) (x y )(z) sup x2s X kx + zk + kx zk " /2 2+"kzk/4 " /2 < 2 " /4. (d))(e). Let x 7! f x be a support functional. By (d) we can coose for ">a so tat for all x,y 2 S X we ave kx y k <",wenever kx + y k > 2. Assume now tat x, y 2 S X wit kx yk <.Ten kf x + f y k 2 (f x + f y )(x + y)

10 8 CHAPTER 4. CONVEXITY AND SMOOTHNESS = f x (x)+f y (y)+ 2 f x(y x)+ 2 f y(x y) 2 kx yk 2, wic implies tat kf x (e))(a). Clear. f y k <", wic proves our claim. Teorem Every uniformly convex and every uniformly smoot Banac space is reflexive. Proof. Assume tat X is uniformly convex, and let x 2 S X. Since B X is w -dense in B X we can find a net (x i ) i2i wic converges wit respect to w to x. Since for every > 0tereisax 2 S X wit lim i2i x (x i )=x (x ) >, it follows tat lim i2i kx i k = and we can terefore assume tat kx i k =, i 2 I. We claim tat (x i ) is a Caucy net wit respect to te norm to x, wic would finis our proof. So let ">0and coose so tat kx+yk > 2 implies tat kx yk <", for any x, y 2 S X. Ten coose x 2 S X, so tat x (x ) > /4, and finally let i 0 2 I so tat x (x i ) /2, for all i i 0. It follows tat kx i + x j k x (x i + x j ) 2 wenever i, j i 0, and tus kx i x j k <", wic verifies our claim. If X is uniformly smoot it follows from Teorem tat X is uniformly convex. Te first part yields tat X is reflexive, wic implies tat X is reflexive. Exercises. Sow tat for tere is a constant c>0 so tat for all ">0, `2(") c" 2. (Here `2 is te modulus of uniform convexity of `2). 2. Prove tat for every ">0, C>and any n 2 N tere is an N = (n, ", C) so tat te following olds: If X is an N dimensional space wic is C-isomorpic to `N ten X as an n-dimensional subspace Y wic is ( + ") ismorpic to `n.

11 4.2. UNIFORM CONVEXITY AND UNIFORM SMOOTHNESS 9 Hint: prove first te following: Assume tat is a norm on `n2 so tat C x < for all x 2 `n2, ten tere is a -normalized block sequence (x,x 2,...x n ) so tat : p C nx b i i= nx b i x i i= nx b i. i= 3. Sow tat n= `n `2 does not admit a norm wic is uniformly convex.