2364 CHAPTER 69. GELFAND TRIPLES
Proof: From Theorem 69.2.9
∫ b
af (t)φ
′ (t)dt
=∫ b
a
(f (a)+
∫ t
af ′ (s)ds
)φ′ (t)dt
= f (a)(φ (b)−φ (a))+∫ b
a
∫ t
af ′ (s)dsφ
′ (t)dt
= f (a)(φ (b)−φ (a))+∫ b
af ′ (s)
∫ b
sφ′ (t)dtds
= f (a)(φ (b)−φ (a))+∫ b
af ′ (s)(φ (b)−φ (s))ds
= f (a)(φ (b)−φ (a))−∫ b
af ′ (s)φ (s)ds+( f (b)− f (a))φ (b)
= f (b)φ (b)− f (a)φ (a)−∫ b
af ′ (s)φ (s)ds.
The interchange in order of integration is justified as in the proof of Theorem 69.2.9.
With this integration by parts formula, the following interesting lemma is obtained.This lemma shows why it was appropriate to define f as in Definition 69.2.2.
Lemma 69.2.11 Let f be given in Definition 69.2.2 and suppose f , f ′ ∈ L1 (a,b;X) . Thenf , f ′ ∈ L1 (2a−b,2b−a;X) also and
f ′ (t)≡
f ′ (t) if t ∈ [a,b]− f ′ (2a− t) if t ∈ [2a−b,a]− f ′ (2b− t) if t ∈ [b,2b−a]
(69.2.8)
Proof: It is clear from the definition of f that f ∈ L1 (2a−b,2b−a;X) and that in fact
∣∣∣∣ f ∣∣∣∣L1(2a−b,2b−a;X)≤ 3 || f ||L1(a,b;X) . (69.2.9)