DIRECT SUM AND DIRECT INTEGRAL OF HILBERT SPACES

1. Direct Sum

Let H1 and H2 be two complex Hilbert spaces. On H1× H_{2}, we define
h(h_{1}, h2), (k1, k2)i = hh1, k1iH_{1}+ hh2, k2iH_{2}.

Here h1, k1 ∈ H_{1} and h2, k2 ∈ H_{2}. Then we can verify that H1 × H_{2} together with the
sesqulinear form defined above is a Hilbert space. We denote this Hilbert space by H_{1}⊕ H_{2}.
Similarly, given a finite sequence of Hilbert spaces H_{1}, · · · , H_{n}, we can define their direct
sum H1⊕ · · · ⊕ H_{n}.

Let {Hn: n ≥ 1} be a sequence of Hilbert spaces over C. Let H be the space of functions ϕ : N →S

nH_{n} so that
(1) ϕ(n) ∈ Hn for each n,
(2) P∞

n=1kϕ(n)k^{2}_{n}< ∞.

The vector space structure of H is defined by

(aψ)(n) = aψ(n), (ψ + ϕ)(n) = ψ(n) + ϕ(n)

for a ∈ C and ψ, ϕ ∈ H. The space of functions H can be equipped with an inner product by

(1.1) hϕ, ψi =

∞

X

n=1

hϕ(n), ψ(n)i_{H}_{n}.

Theorem 1.1. H together with the inner product defined in (1.1) is a Hilbert space.

Proof. It is routine to check that (1.1) is an inner product on H.

Let {ψ_{k}} be Cauchy sequence in H. For each n ≥ 1,

kψ_{k}(n) − ψ_{l}(n)kH_{n} ≤ kψ_{k}− ψ_{l}k_{H}

implies that (ψ_{k}(n)) is a Cauchy sequence in H_{n}. Since H_{n} is complete, we can choose
ψ(n) so that ψ(n) = lim_{k→∞}ψ_{k}(n). We obtain a function ψ : N →S

nH_{n}. Since a Cauchy
sequence in a normed space is bounded, we can choose M ≥ 0 so that kψ_{k}k_{H} ≤ M for all
k. In other words,P∞

n=1kψ_{k}(n)k^{2} ≤ M for all k. Taking lim sup_{k→∞} of this inequality, we
obtain P∞

n=1kψ(n)k^{2}≤ M. This implies that ψ ∈ H.

For any > 0, choose N so that for any k, l ≥ N, kψ_{k}− ψ_{l}k_{H} < /2. Again, taking
lim sup_{l→∞}, we find kψ − ψkkH≤ /2 < for all k ≥ N. This implies that

k→∞lim ψk= ψ.

Hence the sequence (ψk) in H is convergent to ψ in H. We complete the proof of our

assertion.

We will denote H by L∞

n=1H_{n} and call H the direct sum of {Hn}. In this case, we
identify H_{n} with the closed subspace of H consisting of functions ϕ ∈ H so that ϕ(m) = 0
for m 6= n. Then we find that Hn⊥ H_{m} for n 6= m.

1

2 DIRECT SUM AND DIRECT INTEGRAL OF HILBERT SPACES

If H is a Hilbert space and {Hn} is a sequence of closed subspaces so that H_{n}⊥ H_{m} and
for each h ∈ H, there is a unique h_{n}∈ H_{n}for all n so that h =P∞

n=1h_{n}. We can identify H
withL∞

n=1H_{n} via h 7→ ϕh, where ϕh(n) = hn. In this case, we also denote H byL∞
n=1H_{n}.
2. Direct Integral

Let X be a σ-compact^{1}locally compact Hausdorff space and µ is a complete Borel measure
on X. Let {Hx : x ∈ X} be a family of Hilbert spaces parametrized by (points of) X. Let
H be the space of functions f : X →S

x∈XH_{x} so that
(1) f (x) ∈ Hx for each x ∈ X

(2) the function x 7→ hf (x), g(x)iHx is µ-integrable,

(3) if h_{x} ∈ H_{x} for all x ∈ X and x 7→ hh_{x}, g(x)i is integrable for each g ∈ H, then there
is f ∈ H so that f (x) = hx for almost all x ∈ X.

On H, we define

hf, gi = Z

X

hf (x), g(x)i_{x}dµ(x).

Theorem 2.1. The space H together with the inner product defined above is a Hilbert space.

Proof. The proof is similar as above.

We denote H by

H =
Z ^{L}

X

H_{x}dµ(x)
and call H the direct integral of {Hx} over (X, µ).

1A space is σ-compact if it is a countable union of compact sets.