1182 CHAPTER 34. GELFAND TRIPLES AND RELATED STUFF
Consider ⟨Bgp−B
(k
∑i=1
⟨Bgp,ei
⟩ei
),gp−
k
∑i=1
⟨Bgp,ei
⟩ei
⟩If p ∈ (nk,nk+1) , then the above equals zero which implies
Bgp =k
∑i=1
⟨Bgp,ei
⟩Bei
On the other hand, suppose gp = gnk+1 for some nk+1 and so, from the construction, gnk+1 =gp ∈ span(e1, · · · ,ek+1) and therefore,
gp =k+1
∑j=1
a je j
which requires easily that
Bgp =k+1
∑i=1
⟨Bgp,ei
⟩Bei,
the above holding for all k large enough. To see this last claim, note that the coefficients ofBg=∑
mj=1 a jBe j are required to be a j =
⟨Bg,e j
⟩and from the construction,
⟨Bei,e j
⟩= δ i j.
Thus if the upper limit is increased beyond what is needed, the new terms are all zero. Itfollows that for any x ∈ span({gk}∞
k=1) , (finite linear combination of vectors in {gk}∞
k=1)
Bx =∞
∑i=1⟨Bx,ei⟩Bei (34.4.18)
because for all k large enough,
Bx =k
∑i=1⟨Bx,ei⟩Bei
Also note that for such x ∈ span({
g j}∞
j=1
),
⟨Bx,x⟩ =
⟨k
∑i=1⟨Bx,ei⟩Bei,x
⟩=
k
∑i=1⟨Bx,ei⟩⟨Bx,ei⟩
=k
∑i=1|⟨Bx,ei⟩|2 =
∞
∑i=1|⟨Bx,ei⟩|2
Now for x arbitrary, let xk→ x in W where xk ∈ span({gk}∞
k=1) . Then by Fatou’s lemma,
∞
∑i=1|⟨Bx,ei⟩|2 ≤ lim inf
k→∞
∞
∑i=1|⟨Bxk,ei⟩|2
= lim infk→∞
⟨Bxk,xk⟩= ⟨Bx,x⟩ (34.4.19)
≤ ∥Bx∥W ′ ∥x∥W ≤ ∥B∥∥x∥2W