69.4. THE IMPLICIT CASE 2377

which requires easily that

Bgp =k

∑i=1

⟨Bgp,ei

⟩Bei,

the above holding for all k large enough. It follows that for any x ∈ span({gk}∞

k=1) , (finitelinear combination of vectors in {gk}∞

k=1)

Bx =∞

∑i=1⟨Bx,ei⟩Bei (69.4.19)

because for all k large enough,

Bx =k

∑i=1⟨Bx,ei⟩Bei

Also note that for such x ∈ span({gk}∞

k=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⟩ (69.4.20)

≤ ∥Bx∥W ′ ∥x∥W ≤ ∥B∥∥x∥2W

Thus the series on the left converges. Then also, from the above inequality,∣∣∣∣∣⟨

q

∑i=p⟨Bx,ei⟩Bei,y

⟩∣∣∣∣∣≤ q

∑i=p|⟨Bx,ei⟩| |⟨Bei,y⟩|

≤

(q

∑i=p|⟨Bx,ei⟩|2

)1/2( q

∑i=p|⟨By,ei⟩|2

)1/2

≤

(q

∑i=p|⟨Bx,ei⟩|2

)1/2(∞

∑i=1|⟨By,ei⟩|2

)1/2

By 69.4.20,

≤

(q

∑i=p|⟨Bx,ei⟩|2

)1/2(∥B∥∥y∥2

W

)1/2≤

(q

∑i=p|⟨Bx,ei⟩|2

)1/2

∥B∥1/2 ∥y∥W