574 CHAPTER 21. BANACH SPACES

23. ↑Let L : D1→C ([−1,1] ,Z) ,Ly≡ y′. Show L is continuous, defined on D1 and oneto one onto L

(D1). Show L

(D1)= C ([−1,1] ,Z). Thus L−1 is continuous by the

open mapping theorem. Hint: Adapt the Rieman integral to an integral which hasvalues in a Banach space including the fundamental theorem of calculus. Then if u∈C ([−1,1] ,Z) , consider

∫ t0 u(s)ds≡ w(t) and argue Lw = u.

24. ↑Let UZ denote an open set in Z. Let f : UZ → Z be C1. Then define for u ∈D1, f (u)(t)≡ f (u(t)) . Show that f

(D1)⊆C ([−1,1] ,Z) . If UZ consists of u∈D1

such that u(t) ∈UZ for each t ∈ [−1,1] , show that UZ is an open subset of D1. Iff : U → Z is C1, show that f : UZ →D1 is also C1 and that

D f (u)(v)(t) = D f (u(t))(v(t)) .