1204 CHAPTER 34. GELFAND TRIPLES AND RELATED STUFF

Therefore, this term is dominated by an expression of the form∣∣∣∣∣mk−1

∑j=0

⟨∫ t j+1

t j

Y (r)dr,u(t j+1

)−u(t j)

⟩∣∣∣∣∣=

∣∣∣∣∣mk−1

∑j=0

∫ t j+1

t j

⟨Y (r) ,u

(t j+1

)−u(t j)

⟩dr

∣∣∣∣∣=

∣∣∣∣∣mk−1

∑j=0

∫ t j+1

t j

⟨Y (r) ,u

(t j+1

)⟩−

mk−1

∑j=0

∫ t j+1

t j

⟨Y (r) ,u(t j)

⟩∣∣∣∣∣=

∣∣∣∣∫ tm

0⟨Y (r) ,ur (r)⟩dr−

∫ tm

0

⟨Y (r) ,ul (r)

⟩dr∣∣∣∣

≤∫ T

0

∣∣∣⟨Y (r) ,ur (r)−ul (r)⟩∣∣∣dr

However, both ur and ul converge to u in K = Lp (0,T,V ). Therefore, this term mustconverge to 0. Passing to a limit, it follows that for all t ∈ D, the desired formula holds.Thus, for such t ∈ D,

⟨Bu(t) ,u(t)⟩+∫ t

0

⟨B′u,u

⟩dr = ⟨Bu0,u0⟩+2

∫ t

0⟨Y (r) ,u(r)⟩dr

It remains to verify that this holds for all t /∈ N. Let t ∈ NC \D and let t (k) ∈Pk be thelargest point of Pk which is less than t. Suppose t (m)≤ t (k) so that m≤ k. Then

Bu(t (m)) = Bu0 +∫ t(m)

0Y (s)ds,

a similar formula for u(t (k)) . Thus for t > t (m) ,

Bu(t)−Bu(t (m)) =∫ t

t(m)Y (s)ds

which is the same sort of thing already looked at except that it starts at t (m) rather than at0 and u0 = 0. Therefore,

⟨B(u(t (k))−u(t (m))) ,u(t (k))−u(t (m))⟩

+∫ t(k)

t(m)

⟨B′ (s)(u(s)−u(t (m))) ,u(s)−u(t (m))

⟩= 2

∫ t(k)

t(m)⟨Y (s) ,u(s)−u(t (m))⟩ds

Thus, for m≤ k

limm,k→∞

⟨B(u(t (k))−u(t (m))) ,u(t (k))−u(t (m))⟩= 0 (34.5.37)