Chapter 69

Gelfand TriplesLet H be a separable real Hilbert space and let V ⊆ H be a separable Banach space whichis embedded continuously into H and which is also dense in H. Then identifying H and H ′

you can writeV ⊆ H = H ′ ⊆V ′.

This is called a Gelfand triple. If V is reflexive, you could conclude separability of V fromthe separability of H. However, if V is not reflexive, this might not happen. For example,you could take V = L∞ (0,1) and H = L2 (0,1).

Proposition 69.0.1 Suppose V is reflexive and a subset of H a separable Hilbert spacewith the inclusion map continuous. Suppose also that V is dense in H. Then identifying Hand H ′, it follows that H is dense in V ′ and V is separable.

Proof: If H is not dense in V ′, then by the Hahn Banach theorem, there exists φ∗∗ ∈V ′′

such that φ∗∗ (H) = 0 but φ

∗∗ (φ ∗) ̸= 0 for some φ∗ ∈ V ′ \H. Since V is reflexive there

exists v ∈V such that φ∗∗ = Jv for J the standard mapping from V to V ′′. Thus

φ∗∗ (h)≡ ⟨h,v⟩ ≡ (v,h)H = 0

for all h ∈ H. Therefore, v = 0 and so Jv = 0 = φ∗∗ which contradicts φ

∗∗ (φ ∗) ̸= 0.Therefore, H is dense in V ′. Now by Theorem 21.1.16 which says separability of the dualspace implies separability of the space, it follows V is separable as claimed. This provesthe proposition.

From now on, it is assumed V and V ′ are both separable and that H is dense in V ′. Thisis summarized in the following definition.

Definition 69.0.2 V,H,V ′ will be called a Gelfand triple if V,V ′ are separable, V ⊆H withthe inclusion map continuous, H = H ′, and H = H ′ is dense in V ′.

What about the Borel sets on V and H?

Proposition 69.0.3 Denote by B (X) the Borel sets of X where X is any separable Banachspace. Then

B (X) = σ(X ′).

Here σ (X ′) is the smallest σ algebra such that each φ ∈ X ′ is measurable. Also in thecontext of the above definition, B (V ) = σ (i∗H ′) because H ′ is dense in V ′. Here i∗ isthe restriction to V so that i∗h(v) ≡ h(v) ≡ (h,v)H for all v ∈ V and σ (i∗H ′) denotes thesmallest σ algebra such that i∗h is measurable for each h ∈ H ′.

Proof: By Lemma 21.1.6 there exists a countable subset of the unit ball in X ′

{φ n}∞

n=1 = D′

such that||v||X = sup

{|φ (v)| : φ ∈ D′

}.

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