152 CHAPTER 5. FUNCTIONS ON NORMED LINEAR SPACES

If x = Px, then the minimum value of this on the left occurs when t = 0. Functiondefined on [0,1] has its minimum at t = 0. What does it say about the derivativeof this function at t = 0? Next consider the case that for some x the inequalityRe(z− x,y− x)≤ 0. Explain why this shows x = Py.

10. Using Problem 9 and Problem 8 show the projection map, P onto a closed convexsubset is Lipschitz continuous with Lipschitz constant 1. That is |Px−Py| ≤ |x− y| .

11. Suppose, in an inner product space, you know Re(x,y) . Show that you also knowIm(x,y). That is, give a formula for Im(x,y) in terms of Re(x,y). Hint:

(x, iy) =−i(x,y) =−i(Re(x,y)+ iIm(x,y)) =−iRe(x,y)+ Im(x,y)

while, by definition, (x, iy) = Re(x, iy)+ iIm(x, iy) . Now consider matching real andimaginary parts.

12. Let h > 0 be given and let f (t,x)∈Rn for each x∈Rn. Also let (t,x)→ f (t,x) becontinuous and supt,x ∥f (t,x)∥∞

<C < ∞. Let xh (t) be a solution to the following

xh (t) = x0 +∫ t

0f (s,xh (s−h))ds

where xh (s−h) ≡ x0 if s− h ≤ 0. Explain why there exists a solution. Hint:Consider the intervals [0,h] , [h,2h] and so forth. Next explain why these functions{xh}h>0 are equicontinuous and uniformly bounded. Now use the result of Problem6 to argue that there exists a subsequence, still denoted byxh such that limh→0xh =xin C ([0,T ] ;Rn) as discussed in Problem 5. Use what you learned about the Riemannintegral in single variable advanced calculus to explain why you can pass to a limitand conclude that x(t) = x0 +

∫ t0 f (s,x(s))ds Hint:∥∥∥∥∫ t

0f (s,x(s))ds−

∫ t

0f (s,xh (s−h))ds

∥∥∥∥∞

≤∥∥∥∥∫ t

0f (s,x(s))ds−

∫ t

0f (s,x(s−h))ds

∥∥∥∥∞

+

∥∥∥∥∫ t

0f (s,x(s−h))ds−

∫ t

0f (s,xh (s−h))ds

∥∥∥∥∞

≤∫ T

0∥f (s,x(s))−f (s,x(s−h))∥

∞ds

+∫ T

0∥f (s,x(s−h))−f (s,xh (s−h))∥

∞ds

Now use Problem 2 to verify that x′ = f (t,x) , x(0) = x0. When you have donethis, you will have proved the celebrated Peano existence theorem from ordinarydifferential equations.

13. Let |α| ≡ ∑i α i. Let G denote all finite sums of functions of the form p(x)e−a|x|2

where p(x) is a polynomial and a > 0. If you consider all real valued continu-ous functions defined on the closed ball B(0,R) show that if f is such a function,

15210.11.12.13.CHAPTER 5. FUNCTIONS ON NORMED LINEAR SPACESIf x = Px, then the minimum value of this on the left occurs when t = 0. Functiondefined on [0,1] has its minimum at t = 0. What does it say about the derivativeof this function at t = 0? Next consider the case that for some x the inequalityRe(z—x,y—x) <0. Explain why this shows x = Py.Using Problem 9 and Problem 8 show the projection map, P onto a closed convexsubset is Lipschitz continuous with Lipschitz constant 1. That is |Px — Py| < |x—y|.Suppose, in an inner product space, you know Re (x,y). Show that you also knowIm (x,y). That is, give a formula for Im (x,y) in terms of Re (x,y). Hint:(x, iy) = —i(x,y) = —i(Re (x,y) + im (x, y)) = —iRe (x,y) +Im (x,y)while, by definition, (x, iy) = Re (x, iy) + im (x, iy). Now consider matching real andimaginary parts.Let h > 0 be given and let f (t, a”) € R” for each x € R". Also let (t,2) — f (t,x) becontinuous and sup, , || f (t,z)||,, < C < ce. Let a, (t) be a solution to the followingIl-.xy, (t) = 29+ [ f(s,0n(s—A))aswhere a, (s—h) = x if s—h <0. Explain why there exists a solution. Hint:Consider the intervals [0,h] ,[4,2h] and so forth. Next explain why these functions{a@n}p+9 are equicontinuous and uniformly bounded. Now use the result of Problem6 to argue that there exists a subsequence, still denoted by a, such that limy_.9 7, = xin C((0, 7]; IR”) as discussed in Problem 5. Use what you learned about the Riemannintegral in single variable advanced calculus to explain why you can pass to a limitand conclude that x (t) = x9 + Jj f (s,a(s))ds Hint:| [ Fo.e oar [[F o.2n(-—mascoco< | [ Fo.eiopas— [Fels+T+ [IP 6.2(s-M) -F(s.0n(s—M)))ladsNow use Problem 2 to verify that x’ = f (t,2), 2(0) = ap. When you have donethis, you will have proved the celebrated Peano existence theorem from ordinarydifferential equations.Let |a| = ¥;a;. Let Y denote all finite sums of functions of the form p(x) eae”where p(a) is a polynomial and a > 0. If you consider all real valued continu-ous functions defined on the closed ball B(0,R) show that if f is such a function,