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

Let $\mathbb{R}$ be the ordered field of real numbers, with the notation of that item, let $\mathbb{N}$ be the set of natural numbers, and let $(H,V,A)$ be a Hilbert triple, with the inner products, norms and distances of $H$ and $V$ written with the subscripts $H$ and $V$, with $D(A)$ and $A$ as in Hilbert Triple: a Densely and Continuously Embedded Hilbert Space and Its Form Operator §operator, and with zero vector $0_{H}$, which is also the zero vector of $V$ by claim 1 of A Linear Subspace is a Vector Space and Inherits an Inner Product. Convergence of a sequence in $H$ or in $V$ means convergence in $(H,d_{H})$ or in $(V,d_{V})$, and weak convergence in $H$ or in $V$ refers to the respective inner product. Then the following hold.

1. (The embedding)¶ The inclusion map $\iota:V\to H$, $\iota(x)=x$, is a bounded linear map with operator norm at most $1$. A sequence in $V$ converging to $x\in V$ in $V$ converges to $x$ in $H$; and a sequence in $V$ that is Cauchy in $(V,d_{V})$ is Cauchy in $(H,d_{H})$.

2. (The Riesz map)¶ For every $z\in H$ the map $y\mapsto\langle z,y\rangle_{H}$ on $V$ is a bounded linear functional on $V$ with norm at most $|z|_{H}$, so by The Riesz Representation Theorem for a Real Hilbert Space §existence and The Riesz Representation Theorem for a Real Hilbert Space §uniqueness there is a unique $Jz\in V$ with $\langle Jz,y\rangle_{V}=\langle z,y\rangle_{H}$ for every $y\in V$. The map $J:H\to V$ so defined is linear, satisfies $|Jz|_{V}\le|z|_{H}$ (by The Riesz Representation Theorem for a Real Hilbert Space §norm) and $\langle Jz,z'\rangle_{H}=\langle Jz,Jz'\rangle_{V}=\langle z,Jz'\rangle_{H}$ for all $z,z'\in H$, and is injective. Moreover a sequence in $V$ converging weakly to $x\in V$ in $V$ converges weakly to $x$ in $H$.

3. (Domain and range)¶ $D(A)=\{Jz:z\in H\}$, and $A(Jz)=z$ for every $z\in H$ and $J(Ax)=x$ for every $x\in D(A)$. Consequently $D(A)$ is a linear subspace of $V$ and $A:D(A)\to H$ is linear.

4. (Symmetry and coercivity)¶ For $x\in D(A)$ and $y\in V$, $\langle Ax,y\rangle_{H}=\langle x,y\rangle_{V}$; for $x,y\in D(A)$, $\langle Ax,y\rangle_{H}=\langle x,Ay\rangle_{H}$; and for $x\in D(A)$, $\langle Ax,x\rangle_{H}=|x|_{V}^{2}\ge|x|_{H}^{2}$.

5. (Separability)¶ If $(V,d_{V})$ is separable, then so is $(H,d_{H})$.