1202 CHAPTER 34. GELFAND TRIPLES AND RELATED STUFF

≤ ⟨Bu0,u0⟩+2∫ T

0∥Y (r)∥V ′ ∥u

rk (r)∥V dr+ ε

≤ ⟨Bu0,u0⟩+2(∫ T

0∥Y (r)∥p′

V ′ dr)1/p′(∫ T

0∥ur

k (r)∥pV dr

)1/p

+ ε

≤ C (∥Y∥K′ ,∥u∥K)+ ε

whenever k is large enough 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⟨B(t)u(t) ,u(t)⟩+

∫ T

0

⟨B′u,u

⟩ds <C (∥Y∥K′ ,∥u∥K)+ ε

However, ε was arbitrary and the partitions are nested. Hence the above holds for all ε andso

supt∈D⟨B(t)u(t) ,u(t)⟩+

∫ T

0

⟨B′u,u

⟩ds <C (∥Y∥K′ ,∥u∥K)

By 34.5.29 and the integral equation, there is a set of measure zero including all theearlier sets of measure zero N such that for t /∈ N,ul

n (t) ,urn (t)→ u(t) pointwise in V. Also,

B(t)urn (t)→ Bu(t) in V ′. This last can be obtained from the integral equation solved. t→

Bu(t) is continuous into V ′. Then let t /∈ N. We have urn (t)→ u(t) in V . Now B(t)ur

n (t) =B(t)u(sn) where sn ∈ D and sn→ t. Then Bu(t) = B(t)u(t) and

∥B(sn)u(sn)−B(t)u(t)∥V ′ ≤ ∥(B(sn)−B(t))u(sn)∥V ′ +∥B(t)(u(sn)−u(t))∥V ′

≤Ct ∥B(sn)−B(t)∥+C∥u(sn)−u(t)∥Vwhere Ct is a constant which comes because u(sn)→ u(t) in V and so is bounded. Theconstant C is just maxt∈[0,T ] ∥B(t)∥. Then, since the two terms on the right converge to 0 asn→∞, it follows that as sn→ t,B(sn)u(sn)→ B(t)u(t) = Bu(t) in V ′ while u(sn)→ u(t)in V . It follows that for t /∈ N,

⟨Bu(t) ,u(t)⟩+∫ T

0

⟨B′u,u

⟩ds = lim

n→∞⟨Bu(sn) ,u(sn)⟩+

∫ T

0

⟨B′u,u

⟩ds≤C (∥Y∥K′ ,∥u∥K)

Hence,

supt /∈N⟨Bu(t) ,u(t)⟩+

∫ T

0

⟨B′u,u

⟩ds≤C (∥Y∥K′ ,∥u∥K)

It only remains to verify the claim about weak continuity.Consider now the claim that t→ Bu(t) is weakly continuous on NC. Letting v ∈V,s ∈

NC,limt→s⟨Bu(t) ,v⟩= ⟨Bu(s) ,v⟩= ⟨Bu(s) ,v⟩ (34.5.34)