Proof of Elementary Properties of a Hilbert Triple: the Embedding, the Riesz Map and the Form Operator

We use Hilbert Triple: a Densely and Continuously Embedded Hilbert Space and Its Form Operator, the identities of Elementary Identities in a Real Inner Product Space, The Cauchy–Schwarz Inequality in a Real Inner Product Space, Elementary Properties of Bounded Linear Maps and Functionals on Real Inner Product Spaces and The Riesz Representation Theorem for a Real Hilbert Space. Condition (a) of the triple is $|x|_{H}\le|x|_{V}$ for $x\in V$, and condition (b) that $V$ is dense in $H$.

Claim 1. $\iota$ is linear since the operations of $V$ are those of $H$ (claim 1 of A Linear Subspace is a Vector Space and Inherits an Inner Product), and $|\iota x|_{H}=|x|_{H}\le 1\cdot|x|_{V}$, so $\iota$ is bounded with operator norm at most $1$ by Elementary Properties of Bounded Linear Maps and Functionals on Real Inner Product Spaces (i). If $(x_{m})$ in $V$ converges to $x$ in $V$, then for every $\varepsilon>0$ there is $N$ with $|x_{m}-x|_{V}<\varepsilon$ for $m\ge N$, hence $|x_{m}-x|_{H}\le|x_{m}-x|_{V}<\varepsilon$; so $(x_{m})$ converges to $x$ in $H$ (Convergent Sequence in a Metric Space). The Cauchy statement is identical with $|x_{m}-x_{n}|$ in place of $|x_{m}-x|$ (Cauchy Sequence in a Metric Space).

Claim 2. Let $z\in H$ and $\ell_{z}(y)=\langle z,y\rangle_{H}$ for $y\in V$. It is linear by Elementary Identities in a Real Inner Product Space (i), and $|\ell_{z}(y)|\le|z|_{H}|y|_{H}\le|z|_{H}|y|_{V}$ by Cauchy-Schwarz, condition (a) and claim 5 of Elementary Arithmetic in an Ordered Field; so $\ell_{z}$ is a bounded linear functional on $V$ and $\lVert\ell_{z}\rVert\le|z|_{H}$ by Elementary Properties of Bounded Linear Maps and Functionals on Real Inner Product Spaces (i) (via Elementary Properties of Bounded Linear Maps and Functionals on Real Inner Product Spaces (viii)). Since $V$ is a real Hilbert space, The Riesz Representation Theorem for a Real Hilbert Space (i) and The Riesz Representation Theorem for a Real Hilbert Space (ii) give a unique $Jz\in V$ with $\langle y,Jz\rangle_{V}=\ell_{z}(y)$, that is (by symmetry) $\langle Jz,y\rangle_{V}=\langle z,y\rangle_{H}$, for every $y\in V$; and $|Jz|_{V}=\lVert\ell_{z}\rVert\le|z|_{H}$ by The Riesz Representation Theorem for a Real Hilbert Space (iii).

Linearity. For $z,z'\in H$, $\lambda\in\mathbb{R}$ and $y\in V$: $\langle Jz+Jz',y\rangle_{V}=\langle z,y\rangle_{H}+\langle z',y\rangle_{H}=\langle z+z',y\rangle_{H}=\langle J(z+z'),y\rangle_{V}$, so $J(z+z')=Jz+Jz'$ by the uniqueness clause; likewise $J(\lambda z)=\lambda Jz$.

Identities. $\langle Jz,Jz'\rangle_{V}=\langle z,Jz'\rangle_{H}$ by the defining identity of $Jz$ with $y=Jz'\in V$; and $\langle Jz,z'\rangle_{H}=\langle z',Jz\rangle_{H}=\langle Jz',Jz\rangle_{V}=\langle Jz,Jz'\rangle_{V}$ by symmetry and the defining identity of $Jz'$ with $y=Jz$.

Injectivity. If $Jz=0_{H}$ (the zero vector of $V$ by claim 1 of A Linear Subspace is a Vector Space and Inherits an Inner Product), then $\langle z,y\rangle_{H}=\langle Jz,y\rangle_{V}=\langle 0_{H},y\rangle_{V}=0$ for every $y\in V$ (Elementary Identities in a Real Inner Product Space (ii)), so $z\in V^{\perp}$, which is $\{0_{H}\}$ by Orthogonal Projection onto a Closed Linear Subspace of a Real Hilbert Space (vi) and condition (b); hence $z=0_{H}$, and $J$ being linear, it is injective.

Weak convergence. Let $(x_{m})$ in $V$ converge weakly to $x\in V$ in $V$, and let $z\in H$. Then $\langle x_{m},z\rangle_{H}=\langle z,x_{m}\rangle_{H}=\langle Jz,x_{m}\rangle_{V}=\langle x_{m},Jz\rangle_{V}$, which converges to $\langle x,Jz\rangle_{V}=\langle x,z\rangle_{H}$ by Weak Convergence of a Sequence in a Real Inner Product Space applied with the vector $Jz\in V$. As $z$ was arbitrary, $(x_{m})$ converges weakly to $x$ in $H$.

Claim 3. If $x\in D(A)$ and $z=Ax$, then $\langle x,y\rangle_{V}=\langle z,y\rangle_{H}=\langle Jz,y\rangle_{V}$ for every $y\in V$, so $x=Jz=J(Ax)$ by the uniqueness in claim 2 (or by Elementary Identities in a Real Inner Product Space (iii) with $y=x-Jz$). Conversely, for $z\in H$ the vector $x=Jz$ satisfies $\langle x,y\rangle_{V}=\langle z,y\rangle_{H}$ for all $y\in V$, so $x\in D(A)$ with $Ax=z$ by Hilbert Triple: a Densely and Continuously Embedded Hilbert Space and Its Form Operator (iv); that is, $A(Jz)=z$. Hence $D(A)=J(H)$. Since $J$ is linear, $J(H)$ contains $0_{H}=J0_{H}$ and is closed under sums and scalar multiples, so $D(A)$ is a linear subspace of $V$; and $A(Jz+Jz')=A(J(z+z'))=z+z'=A(Jz)+A(Jz')$, $A(\lambda Jz)=A(J(\lambda z))=\lambda z=\lambda A(Jz)$, so $A$ is linear.

Claim 4. The first identity is Hilbert Triple: a Densely and Continuously Embedded Hilbert Space and Its Form Operator (iv). For $x,y\in D(A)$, by it and symmetry, $\langle Ax,y\rangle_{H}=\langle x,y\rangle_{V}=\langle y,x\rangle_{V}=\langle Ay,x\rangle_{H}=\langle x,Ay\rangle_{H}$. With $y=x$, $\langle Ax,x\rangle_{H}=\langle x,x\rangle_{V}=|x|_{V}^{2}\ge|x|_{H}^{2}$ by condition (a) and claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field.

Claim 5. Let $D\subseteq V$ be countable and dense in $(V,d_{V})$, and let $\bar{D}$ be its closure in $(H,d_{H})$, a closed subset of $H$ containing $D$ by claims 1 and 2 of The Closure is the Smallest Closed Superset. If $x\in V$, then by Sequential Characterization of the Closure in a Metric Space (in $V$) there is a sequence in $D$ converging to $x$ in $V$, hence in $H$ by claim 1, so $x\in\bar{D}$ by the same lemma in $H$. Thus $V\subseteq\bar{D}$, and by claim 3 of The Closure is the Smallest Closed Superset the closure of $V$ in $H$, which is $H$ by condition (b), is contained in $\bar{D}$. So $\bar{D}=H$: $D$ is a countable dense subset of $(H,d_{H})$, which is therefore separable.