Hilbert Triples: Standing Notation and Background

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

1. (The triple)(i) $H$ 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 (i), Hilbert Triple: a Densely and Continuously Embedded Hilbert Space and Its Form Operator (ii), Hilbert Triple: a Densely and Continuously Embedded Hilbert Space and Its Form Operator (iii)). Their inner products, norms and distances carry the subscripts $H$ and $V$; the unsubscripted notation of Real Hilbert Spaces: Standing Notation and Background (ii) 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)(ii) 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 (v).

3. (The form operator, the Riesz map and the resolvent)(iii) $D(A)\subseteq V$ 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 (ii), 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 (i), 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 (ii).

4. (The penalty function)(iv) $h:V\to\mathbb{R}$ denotes the function $h(x)=\tfrac12|x|_{V}^{2}$, where $\tfrac12$ is $2^{-1}$; it is called the penalty function of the triple.

5. (Open subsets of $H$ and their traces)(v) For a subset $U$ of $H$ that is open in $H$, $W=D(A)\cap U$ denotes the trace of the domain of the form operator on $U$. The sets $U$, $V\cap U$ and $W$ are metric spaces under the restriction of $d_{H}$, as in Real Hilbert Spaces: Standing Notation and Background (iii), written $(U,d_{H})$, $(V\cap U,d_{H})$ and $(W,d_{H})$. For a function on $U$, on $V\cap U$ or on $W$, semicontinuity and local extrema refer to that set as a subset of the metric space $(H,d_{H})$, and so do, when the set is nonempty, being bounded above or below near each point and the semicontinuous envelopes; having closed superlevel sets in $(U,d_{H})$ refers to the metric space $(U,d_{H})$. Continuity, uniform continuity and Lipschitz continuity of a function on $U$, on $V\cap U$ or on $W$ likewise refer to that set as a subset of the metric space $(H,d_{H})$. The space $H$ is itself open in $H$ by Real Hilbert Spaces: Standing Notation and Background (iii) and so may be taken as $U$; in that case $V\cap U$ is $V$, carrying the metric $d_{H}$ and not $d_{V}$, and $W$ is $D(A)$. Differentiability, gradients, Hessians and the classes $C^{1}(U)$ and $C^{2}(U)$ of functions on $U$ are those of Real Hilbert Spaces: Series, Products, Orthonormal Bases and Differential Calculus (iv) for the space $H$, so that $D\varphi(x)\in H$ and $D^{2}\varphi(x)\in\mathrm{Sym}(H)$ for $\varphi\in C^{2}(U)$ and $x\in U$.

6. (Forms on $H$ and on $V$, and restriction to $V$)(vi) $\mathrm{Sym}(H)$ and $\mathrm{Sym}(V)$ are the sets of bounded symmetric bilinear forms on $H$ and on $V$, each with the norm $\lVert\cdot\rVert$, the order $\preceq$, the sums, differences and scalar multiples of forms, the zero form $0_{\mathrm{Sym}}$, and the identity form and its multiples, all as fixed in Real Hilbert Spaces: Standing Notation and Background (v) for the space in question; the identity forms are written $I_{H}$ and $I_{V}$, so that $I_{V}(x,y)=\langle x,y\rangle_{V}$ for $x,y\in V$, and when the two norms must be told apart that of $\mathrm{Sym}(V)$ is written $\lVert\cdot\rVert_{\mathrm{Sym}(V)}$. For $Y\in\mathrm{Sym}(H)$, $Y|_{V}\in\mathrm{Sym}(V)$ denotes its restriction to $V$, which is legitimate because $|x|_{H}\le|x|_{V}$ for $x\in V$ by clause 1.

7. (Background)(vii) 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.