Let H be the space of functions ϕ : N →S nHn so that (1) ϕ(n

DIRECT SUM AND DIRECT INTEGRAL OF HILBERT SPACES

1. Direct Sum

Let H1 and H2 be two complex Hilbert spaces. On H1× H₂, we define h(h₁, h2), (k1, k2)i = hh1, k1iH₁+ hh2, k2iH₂.

Here h1, k1 ∈ H₁ and h2, k2 ∈ H₂. Then we can verify that H1 × H₂ together with the sesqulinear form defined above is a Hilbert space. We denote this Hilbert space by H₁⊕ H₂. Similarly, given a finite sequence of Hilbert spaces H₁, · · · , 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)k²_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_Hn.

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_lk_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_kk_H ≤ M for all k. In other words,P∞

n=1kψ_k(n)k² ≤ M for all k. Taking lim sup_k→∞ of this inequality, we obtain P∞

n=1kψ(n)k²≤ M. This implies that ψ ∈ H.

For any  > 0, choose N so that for any k, l ≥ N, kψ_k− ψ_lk_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_nfor 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-compact¹locally 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_xdµ(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_xdµ(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.