1188 CHAPTER 34. GELFAND TRIPLES AND RELATED STUFF
Lemma 34.4.5 In the above situation,
supt∈NC⟨Bu(t) ,u(t)⟩ ≤C (∥Y∥K′ ,∥u∥K)
Also, t→Bu(t) is weakly continuous with values in W ′ on NC where N is the set of measurezero where Bu(t) ̸= B(u(t)).
Proof: From the above formula of Lemma 34.4.4 applied to the kth partition of [0,T ]described above,
⟨Bu(tm) ,u(tm)⟩−⟨Bu0,u0⟩=m−1
∑j=0
⟨Bu(t j+1
),u(t j+1
)⟩−⟨Bu(t j) ,u(t j)
⟩
=m−1
∑j=0
2∫ t j+1
t j
⟨Y (r) ,u
(t j+1
)⟩dr−
⟨B(u(t j+1
)−u(t j)
),u(t j+1
)−u(t j)
⟩=
m−1
∑j=0
2∫ t j+1
t j
⟨Y (r) ,urk (r)⟩dr−
⟨B(u(t j+1
)−u(t j)
),u(t j+1
)−u(t j)
⟩Thus, discarding the negative terms and denoting by Pk the kth of these partitions,
supt j∈Pk
⟨Bu(t j) ,u(t j)
⟩≤ ⟨Bu0,u0⟩+2
∫ T
0|⟨Y (r) ,ur
k (r)⟩|dr
≤ ⟨Bu0,u0⟩+2∫ T
0∥Y (r)∥V ′ ∥u
rk (r)∥V dr
≤ ⟨Bu0,u0⟩+2(∫ T
0∥Y (r)∥p′
V ′ dr)1/p′(∫ T
0∥ur
k (r)∥pV dr
)1/p
≤ C (∥Y∥K′ ,∥u∥K)
because these partitions are chosen such that
limk→∞
(∫ T
0∥ur
k (r)∥pV
)1/p
=
(∫ T
0∥u(r)∥p
V
)1/p
and so these are bounded. This has shown that for the dense subset of [0,T ] , D≡ ∪kPk,
supt∈D⟨Bu(t) ,u(t)⟩<C (∥Y∥K′ ,∥u∥K)
From Lemma 34.4.2 above, there exists {ei}⊆V such that⟨Bei,e j
⟩= δ i j and for t /∈N,
⟨Bu(t) ,u(t)⟩=∞
∑k=1|⟨Bu(t) ,ei⟩|2 = sup
m
m
∑k=1|⟨Bu(t) ,ei⟩|2