34.2. STANDARD TECHNIQUES IN EVOLUTION EQUATIONS 1165
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 34.2.8, there exists a constant, denoted as f (a) such that
f (t)−∫ t
af ′ (s)ds = f (a)
There is also a useful theorem about continuity of pointwise evaluation.
Corollary 34.2.10 Let f , f ′ ∈ L1 (a,b;X) so that
f (t) = f (0)+∫ t
0f ′ (s)ds (34.2.8)
where in this formula, t → f (t) is the continuous representative of f . Then there exists aconstant C such that for each t ∈ [a,b] ,
∥ f (t)∥X ≤C(∥ f∥L1(a,b;X)+
∥∥ f ′∥∥
L1(a,b;X)
)