43.1. SOME STANDARD TECHNIQUES IN EVOLUTION EQUATIONS 1419

Lemma 43.1.13 Let f be given in Definition 43.1.2 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]

(43.1.1)

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) . (43.1.2)

Let φ ∈C∞c (2a−b,2b−a) . Then from the integration by parts formula,∫ 2b−a

2a−bf (t)φ

′ (t)dt

=∫ b

af (t)φ

′ (t)dt +∫ 2b−a

bf (2b− t)φ

′ (t)dt +∫ a

2a−bf (2a− t)φ

′ (t)dt

=∫ b

af (t)φ

′ (t)dt +∫ b

af (u)φ

′ (2b−u)du+∫ b

af (u)φ

′ (2a−u)du

= f (b)φ (b)− f (a)φ (a)−∫ b

af ′ (t)φ (t)dt− f (b)φ (b)+ f (a)φ (2b−a)

+∫ b

af ′ (u)φ (2b−u)du− f (b)φ (2a−b)

+ f (a)φ (a)+∫ b

af ′ (u)φ (2a−u)du

= −∫ b

af ′ (t)φ (t)dt +

∫ b

af ′ (u)φ (2b−u)du+

∫ b

af ′ (u)φ (2a−u)du

= −∫ b

af ′ (t)φ (t)dt−

∫ 2b−a

b− f ′ (2b− t)φ (t)dt−

∫ a

2a−b− f ′ (2a− t)φ (t)dt

= −∫ 2b−a

2a−bf ′ (t)φ (t)dt

where f ′ (t) is given in 43.1.1.

Definition 43.1.14 Let V be a Banach space and let H be a Hilbert space. (TypicallyH = L2 (Ω)) Suppose V ⊆ H is dense in H meaning that the closure in H of V gives H.Then it is often the case that H is identified with its dual space, and then because of thedensity of V in H, it is possible to write

V ⊆ H = H ′ ⊆V ′

When this is done, H is called a pivot space. Another notation which is often used is ⟨ f ,g⟩to denote f (g) for f ∈V ′ and g ∈V. This may also be written as ⟨ f ,g⟩V ′,V . Another termis that V ⊆ H = H ′ ⊆V ′ is called a Gelfand triple.