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,