34.4. THE IMPLICIT CASE 1181
Proof: To see this, note that if p ∈(n j,n j+1
), then by assumption,⟨
B
(gp−
j
∑i=1
⟨Bgp,ei
⟩ei
),gp−
j
∑i=1
⟨Bgp,ei
⟩ei
⟩= 0
Therefore,
Bgp =j
∑i=1
⟨Bgp,ei
⟩Bei
Also, by assumption, if p > nk⟨B
(gp−
k
∑i=1
⟨Bgp,ei
⟩ei
),gp−
k
∑i=1
⟨Bgp,ei
⟩ei
⟩= 0
so
Bgp =k
∑i=1
⟨Bgp,ei
⟩Bei
which shows that span({
Bg j}∞
j=1
)⊆ span
({Bei}k
i=1
). If ∑
ki=1 ciBei = 0, then for j ≤ k,
0 =k
∑i=1
ci⟨Bei,e j
⟩= c j
so {Bei}ki=1 is a basis for span
({Bg j}∞
j=1
)= B
(span
({g j}∞
j=1
)). Hence if x ∈W, then
letting xr ∈ span({
g j}∞
j=1
)with xr→ x in W, it follows
Bxr =k
∑i=1
aiBei =k
∑i=1⟨Bxr,ei⟩Bei
Then passing to a limit, you get
Bx =k
∑i=1⟨Bx,ei⟩Bei
Thus {Bei}ki=1 is a basis for BW . This proves the claim.
If this happens, the process being described stops. You have found what is desiredwhich has only finitely many vectors involved.
If the process does not stop, let
ek+1 ≡gnk+1 −∑
ki=1⟨Bgnk+1 ,ei
⟩ei⟨
B(gnk+1 −∑
ki=1⟨Bgnk+1 ,ei
⟩ei),gnk+1 −∑
ki=1⟨Bgnk+1 ,ei
⟩ei⟩1/2
Thus, as in the usual argument for the Gram Schmidt process,⟨Bei,e j
⟩= δ i j for i, j≤ k+1.
This is already known for i, j≤ k. Letting l ≤ k, and using the orthogonality already shown,
⟨Bek+1,el⟩ = C
⟨B
(gnk+1 −
k
∑i=1
⟨Bgnk+1 ,ei
⟩ei
),el
⟩= C
(⟨Bgk+1,el⟩−
⟨Bgnk+1 ,el
⟩)= 0