34.6. ANOTHER APPROACH 1213

By choosing a subsequence we can use 34.6.48 to obtain

un→ u a.e. and in V (34.6.49)(i∗B̃un

)′→ (i∗Bu)′ a.e. and in V ′.

From 34.6.49,

Re⟨(i∗B̃un

)′(t) ,un (t)⟩ → Re⟨(i∗Bu)′ (t) ,u(t)⟩ a.e. t ∈ [0,T ] (34.6.50)

⟨B′ (t)un (t) ,un (t)⟩ → ⟨B′ (t)u(t) ,u(t)⟩ a.e. t ∈ [0,T ] . (34.6.51)

If g ∈ L∞ (0,T ) ,

limn→∞

∫ T

0g(t)⟨

(i∗B̃un

)′(t) ,un (t)⟩dt = lim

n→∞⟨(i∗B̃un

)′,gun (t)⟩

= ⟨(i∗Bu)′ ,gu⟩=∫ T

0g(t)⟨(i∗Bu)′ (t) ,u(t)⟩dt.

Thus we have the following weak convergence:

Re⟨(i∗B̃un

)′,un⟩⇀ Re⟨(i∗Bu)′ ,u⟩ in L1 (0,T ) .

Similarly,⟨B′un,un⟩⇀ ⟨B′u,u⟩ in L1 (0,T )

It follows from 34.6.47 that

⟨B̃un,un⟩′ (·) converges a.e. and weakly in L1 (0,T ) .

⟨B̃un,un⟩(·) converges a.e. and strongly in L1 (0,T ) to ⟨Bu,u⟩(·) .

Therefore, ⟨B̃un,un⟩′ (·) converges a.e. and weakly in L1 (0,T ) to ⟨Bu,u⟩′ (·) . Since ⟨B̃u,u⟩and ⟨B̃u,u⟩′ are both in L1 (0,T ) , this proves part 1 in the case where v = u. This alsoestablishes formula 2. To get 1 for u ̸= v, apply what was just shown to

⟨B(t)(u(t)+ v(t)) ,u(t)+ v(t)⟩.

Next let t ∈ [0,T ] and use 34.6.47 to write

⟨B̃un,un⟩(t) =

2Re∫ t

−T⟨(i∗B̃un

)′(s) ,un (s)⟩ds−2Re

∫ t

−T⟨B̃′ (s)un (s) ,un (s)⟩ds. (34.6.52)

Using 34.6.49, we let n→ ∞ in 34.6.52 and obtain

⟨B̃u,u⟩(t) = 2Re∫ t

−T⟨(i∗B̃ũ

)′(s) , ũ(s)⟩ds−2Re

∫ t

−T⟨B̃′ (s) ũ(s) , ũ(s)⟩ds. (34.6.53)

Hence from 34.6.46,

|⟨B̃u,u⟩(t) | ≤C[||(i∗B̃ũ

)′ ||V ′[−T,2T ]

||ũ||V[−T,2T ]+ ||ũ||2V[−T,2T ]

]