Spectral Theory in Hilbert Space

20 3. HILBERT SPACE
ˆvn = limj→∞〈vnj, en〉 exists. I claim that {vjj} ∞ j=1 converges weakly to
v = ˆvnen. Note that {〈vjj, en〉} ∞ j=1 converges to ˆvn since it is a subsequence
of {〈vnj, en〉} ∞ j=1 from j = n on. Furthermore N
n=1 |ˆvn| 2 ≤ C 2
for all N since it is the limit as j → ∞ of N
n=1 |〈vNj, en〉| 2 which
by Bessel’s inequality is bounded by vNj 2 ≤ C 2 . It follows that
∞
n=1 |ˆvn| 2 ≤ C 2 so that v is actually an element of H.
To show the weak convergence, let u = ûnen be arbitrary in
H. Suppose ε > 0 given arbitrarily. Writing u = u ′ + u ′′ where
u ′ = N
n=1 ûnen we may now choose N so large that u ′′ < ε so that
|〈vjj, u ′′ 〉| < Cε. Furthermore |〈v, u ′′ 〉| < Cε and 〈vjj, u ′ 〉 → 〈v, u ′ 〉
so limj→∞|〈vjj, u〉 − 〈v, u〉| ≤ 2Cε. Since ε > 0 is arbitrary the weak
convergence follows.
The converse is an immediate consequence of the Banach-Steinhaus
principle of uniform boundedness.
Theorem 3.10 (Banach-Steinhaus). Let ℓ1, ℓ2, . . . be a sequence of
bounded linear forms on a Banach space B which is pointwise bounded,
i.e., such that for each u ∈ B the sequence ℓ1(u), ℓ2(u), . . . is bounded.
Then ℓ1, ℓ2, . . . is uniformly bounded, i.e., there is a constant C such
that |ℓj(u)| ≤ Cu for every u ∈ B and j = 1, 2, . . . .
Assuming Theorem 3.10 (for a proof, see Appendix A), we can
complete the proof of Theorem 3.9, since a weakly convergent sequence
v1, v2, . . . can be identified with a sequence of linear forms ℓ1, ℓ2, . . .
by setting ℓj(u) = 〈u, vj〉. Since a convergent sequence of numbers
is bounded it follows that we have a pointwise bounded sequence of
linear functionals. By Theorem 3.10 there is a constant C such that
|〈u, vj〉| ≤ Cu for every u ∈ H and j = 1, 2, . . . . In particular,
setting u = vj gives vj ≤ C for every j.