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

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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, · · · , Hn, we can define their direct sum H1⊕ · · · ⊕ Hn.

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

nHn so that (1) ϕ(n) ∈ Hn for each n, (2) P

n=1kϕ(n)k2n< ∞.

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)iHn.

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 {ψk} be Cauchy sequence in H. For each n ≥ 1,

k(n) − ψl(n)kHn ≤ kψk− ψlkH

implies that (ψk(n)) is a Cauchy sequence in Hn. Since Hn is complete, we can choose ψ(n) so that ψ(n) = limk→∞ψk(n). We obtain a function ψ : N →S

nHn. Since a Cauchy sequence in a normed space is bounded, we can choose M ≥ 0 so that kψkkH ≤ M for all k. In other words,P

n=1k(n)k2 ≤ M for all k. Taking lim supk→∞ 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, kψk− ψlkH < /2. Again, taking lim supl→∞, 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=1Hn and call H the direct sum of {Hn}. In this case, we identify Hn 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⊥ Hm for n 6= m.

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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 Hn⊥ Hm and for each h ∈ H, there is a unique hn∈ Hnfor all n so that h =P

n=1hn. We can identify H withL

n=1Hn via h 7→ ϕh, where ϕh(n) = hn. In this case, we also denote H byL n=1Hn. 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∈XHx 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 hx ∈ Hx for all x ∈ X and x 7→ hhx, 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)ixdµ(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

Hxdµ(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.

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