69.2. STANDARD TECHNIQUES IN EVOLUTION EQUATIONS 2363
Proof:∫ b
a
(f (t)−
∫ t
af ′ (s)ds
)φ′ (t)dt =
∫ b
af (t)φ
′ (t)dt−∫ b
a
∫ t
af ′ (s)φ
′ (t)dsdt.
Now consider∫ b
a∫ t
a f ′ (s)φ′ (t)dsdt. Let Λ ∈ X ′. Then it is routine from approximating f ′
with simple functions to verify
Λ
(∫ b
a
∫ t
af ′ (s)φ
′ (t)dsdt)=∫ b
a
∫ t
aΛ(
f ′ (s))
φ′ (t)dsdt.
Now the ordinary Fubini theorem can be applied to obtain
=∫ b
a
∫ b
sΛ(
f ′ (s))
φ′ (t)dtds = Λ
(∫ b
a
∫ b
sf ′ (s)φ
′ (t)dtds).
Since X ′ separates the points of X , it follows
∫ b
a
∫ t
af ′ (s)φ
′ (t)dsdt =∫ b
a
∫ b
sf ′ (s)φ
′ (t)dtds.
Therefore,
∫ b
a
(f (t)−
∫ t
af ′ (s)ds
)φ′ (t)dt
=∫ b
af (t)φ
′ (t)dt−∫ b
a
∫ b
sf ′ (s)φ
′ (t)dtds
=∫ b
af (t)φ
′ (t)dt−∫ b
af ′ (s)
∫ b
sφ′ (t)dtds
=∫ b
af (t)φ
′ (t)dt +∫ b
af ′ (s)φ (s)ds = 0.
Therefore, by Lemma 69.2.8, there exists a constant, denoted as f (a) such that
f (t)−∫ t
af ′ (s)ds = f (a)
The integration by parts formula is also important.
Corollary 69.2.10 Suppose f , f ′ ∈ L1 (a,b;X) and suppose φ ∈ C1 ([a,b]) . Then the fol-lowing integration by parts formula holds.
∫ b
af (t)φ
′ (t)dt = f (b)φ (b)− f (a)φ (a)−∫ b
af ′ (t)φ (t)dt.