Hilbert Triples: Standing Notation and Background

Throughout we work in the setting of Real Hilbert Spaces: Standing Notation and Background, whose notation is in force for each real Hilbert space named below, and we fix a Hilbert triple $(H,V,A)$.

1. (The triple)¶ $H$ and $V$ are the real Hilbert spaces of the triple: $V$ with $\langle\cdot,\cdot\rangle_{V}$ is itself a real Hilbert space, a dense linear subspace of $H$ with $|x|_{H}\le|x|_{V}$ for $x\in V$ (Hilbert Triple: a Densely and Continuously Embedded Hilbert Space and Its Form Operator §triple, Hilbert Triple: a Densely and Continuously Embedded Hilbert Space and Its Form Operator §embedding, Hilbert Triple: a Densely and Continuously Embedded Hilbert Space and Its Form Operator §dense). Their inner products, norms and distances carry the subscripts $H$ and $V$; the unsubscripted notation of Real Hilbert Spaces: Standing Notation and Background §space is not used in this setting. $0_{H}$ is the zero vector of $H$, which is also that of $V$ by claim 1 of A Linear Subspace is a Vector Space and Inherits an Inner Product. Convergence, weak convergence, openness, closedness, density and boundedness in $H$ or in $V$ refer to the respective distance and inner product, and are named with the space (for instance converges in $H$).

2. (Standing hypothesis: separability)¶ The metric space $(V,d_{V})$ is separable; hence so is $(H,d_{H})$, by Elementary Properties of a Hilbert Triple: the Embedding, the Riesz Map and the Form Operator §separable.

3. (The form operator, the Riesz map and the resolvent)¶ $D(A)\subseteq V$ and $A:D(A)\to H$ are the domain and the form operator of the triple, so that $\langle Ax,y\rangle_{H}=\langle x,y\rangle_{V}$ for $x\in D(A)$ and $y\in V$; $J:H\to V$ is the map of Elementary Properties of a Hilbert Triple: the Embedding, the Riesz Map and the Form Operator §riesz-map, called the Riesz map, with $\langle Jz,y\rangle_{V}=\langle z,y\rangle_{H}$ for $z\in H$, $y\in V$. For a real $\alpha>0$, $R_{\alpha}:H\to V$ is the map of The Resolvent of the Form Operator of a Hilbert Triple: Minimisation, Contraction, and Density of the Domain §minimiser, called the resolvent map, whose values lie in $D(A)$ and satisfy $A(R_{\alpha}\bar{x})+2\alpha R_{\alpha}\bar{x}=2\alpha\bar{x}$ by The Resolvent of the Form Operator of a Hilbert Triple: Minimisation, Contraction, and Density of the Domain §euler-lagrange.

4. (Background)¶ The results Elementary Properties of a Hilbert Triple: the Embedding, the Riesz Map and the Form Operator (in particular, convergence and weak convergence in $V$ imply the same in $H$ with the same limit), The Resolvent of the Form Operator of a Hilbert Triple: Minimisation, Contraction, and Density of the Domain and Weak Compactness and Closedness of Bounded Subsets of the Small Space of a Hilbert Triple are in force by reference, the separability hypotheses of the last two being supplied by clause 2.