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Separability and Hilbert structure of the no-slip dual

Statement

Assume Countable Choice and let $V$ be any separable real Hilbert space. Its continuous real-linear dual $V^{*}$ , with norm $∥ ℓ ∥_{V^{*}} = \sup_{∥ v ∥_{V} \leq 1} ∣ ℓ (v) ∣$ , is a separable real Hilbert space. The Riesz map $J : V \to V^{*}, J w (v) = (w, v)_{V}$ is a surjective linear isometry.

Statement references

Berselli and Spirito, On the Existence of Leray–Hopf Weak Solutions, §2.2, pp. 4–5.

Leonid Ryzhik, Math 256B Lecture Notes, §§5.2.2–5.4, pp. 42–48.

Facts & Assumptions

Given: Countable Choice and a separable real Hilbert space $V$ .

[F1]

$V$ is complete and has a countable dense subset, as specified by the real Hilbert and separability hypotheses in the Statement.

[F2]

The Axiom of Countable Choice ( ${A C}_{ω}$ ): from each countable sequence of nonempty sets one may select one element; this selects an approximating sequence for the distance minimization below.

Proof

1.1givenF1F2algebra

Let $ℓ \in V^{*}$ be nonzero. Its kernel $K = {v : ℓ (v) = 0}$ is a closed linear subspace. Choose $x \notin K$ and put $d_{0} = \inf_{k \in K} ∥ x - k ∥_{V}$ . By [F2], choose $k_{n} \in K$ with $∥ x - k_{n} ∥_{V} \to d_{0}$ . Since $(k_{n} + k_{m}) / 2 \in K$ , the parallelogram identity yields $∥ k_{n} - k_{m} ∥_{V}^{2} = 2 ∥ x - k_{n} ∥_{V}^{2} + 2 ∥ x - k_{m} ∥_{V}^{2} - 4 {∥ x - \frac{k_{n} + k_{m}}{2} ∥}_{V}^{2} \leq 2 ∥ x - k_{n} ∥_{V}^{2} + 2 ∥ x - k_{m} ∥_{V}^{2} - 4 d_{0}^{2},$ which tends to zero as $n, m \to \infty$ . Thus $(k_{n})$ is Cauchy; completeness and closedness give a limit $k_{0} \in K$ attaining the distance. Set $y = x - k_{0} \neq 0$ ; because $k_{0} \in K$ and $x \notin K$ , $ℓ (y) = ℓ (x) \neq 0$ . For any $k \in K$ , the quadratic $∥ y - t k ∥_{V}^{2}$ has its minimum at $t = 0$ , so its linear coefficient vanishes and $(y, k)_{V} = 0$ .

2.1givenF1step 1.1algebra

For any $z \in V$ , the vector $z - ℓ (z) y / ℓ (y)$ belongs to $K$ . Orthogonality from step 1.1 therefore gives $ℓ (z) = \frac{ℓ (y)}{∥ y ∥_{V}^{2}} (y, z)_{V} = (w_{ℓ}, z)_{V}, w_{ℓ} : = \frac{ℓ (y)}{∥ y ∥_{V}^{2}} y .$ The zero functional is represented by $w_{0} = 0$ . If two vectors represent the same functional, their difference is orthogonal to every vector and hence is zero. Cauchy–Schwarz gives $∥ J w ∥_{V^{*}} \leq ∥ w ∥_{V}$ , while testing at $v = w / ∥ w ∥_{V}$ for $w \neq 0$ gives equality. Thus $J$ is a linear isometric bijection.

3.1givenF1step 2.1∎

The image under an isometry of a countable dense subset of $V$ is countable and dense in $V^{*}$ because $J$ is onto. The Riesz isometry transports the Hilbert inner product of $V$ to $V^{*}$ , so the dual is a separable real Hilbert space.

Depends on

The Axiom of Countable Choice ( ${A C}_{ω}$ )