Proof of The Penalty Function $h=\tfrac12|\cdot|_V^2$ of a Hilbert Triple: Expansion Identities, Closed Sublevel Sets, Density and Local Bounds

Throughout, $x$ and $y$ denote points of $V$, $\tfrac12$ denotes $2^{-1}$, which is positive by claim 8 of Elementary Order Arithmetic in an Ordered Field, and products of nonnegative real numbers are nonnegative by claim 5 of Elementary Arithmetic in an Ordered Field (multiply $0\le y$ by a nonnegative $a$ and use $a\cdot0=0$). Vector identities such as $y=x+(y-x)$ are those of Elementary Identities in a Vector Space, applied in the vector space $V$, whose operations are the restrictions of those of $H$ (claim 1 of A Linear Subspace is a Vector Space and Inherits an Inner Product, as recorded in Hilbert Triples: Standing Notation and Background (i)).

Claim 1. The norm of $V$ is nonnegative by Real Inner Product Spaces (ii), and $|x|_{H}\le|x|_{V}$ by Hilbert Triples: Standing Notation and Background (i); hence $0\le|x|_{H}\le|x|_{V}$, and claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field gives $0=0^{2}\le|x|_{H}^{2}\le|x|_{V}^{2}$, where $0^{2}=0$ by claim 4 of Properties of Natural Number Powers in a Field. Multiplying by $\tfrac12$ (claim 5 of Elementary Arithmetic in an Ordered Field) yields $0\le\tfrac12|x|_{H}^{2}\le h(x)$. Finally $|0_{H}|_{V}=0$ by Elementary Identities in a Real Inner Product Space (ii), so $h(0_{H})=\tfrac12\cdot0^{2}=0$.

Claim 2. Since $y=x+(y-x)$ with $y-x\in V$, Elementary Identities in a Real Inner Product Space (v) in the inner product space $V$ gives

$$|y|_{V}^{2}=|x|_{V}^{2}+2\langle x,y-x\rangle_{V}+|y-x|_{V}^{2}.$$

Multiplying by $\tfrac12$ and using $\tfrac12\cdot2=1$ gives $h(y)=h(x)+\langle x,y-x\rangle_{V}+h(y-x)$.

Claim 3. Let $x\in D(A)$ and $y\in V$. Then $y-x\in V$, so $\langle x,y-x\rangle_{V}=\langle Ax,y-x\rangle_{H}$ by Hilbert Triple: a Densely and Continuously Embedded Hilbert Space and Its Form Operator (iv), and claim 2 becomes $h(y)=h(x)+\langle Ax,y-x\rangle_{H}+h(y-x)$.

Claim 4. We apply Functions with Closed Superlevel Sets: Sequential Characterisation, Semicontinuity, Perturbation and Limits (i) to the function $-h$ on the subset $V$ of the metric space $(H,d_{H})$. Accordingly, let $(x_{m})_{m\in\mathbb{N}}$ be a sequence in $V$ converging in $H$ to a point $x\in H$, and let $t\in\mathbb{R}$ satisfy $t\le-h(x_{m})$ for every $m$; we must show $x\in V$ and $t\le-h(x)$. Put $c=-t$, so that $h(x_{m})\le c$ for every $m$ (claim 4 of Elementary Order Arithmetic in an Ordered Field); since $0\le h(x_{m})$ by claim 1, also $0\le c$ and $0\le 2c$.

First, $|x_{m}|_{V}\le1+2c$ for every $m$. Indeed, if $|x_{m}|_{V}\le1$ this follows from $1\le1+2c$. Otherwise $1<|x_{m}|_{V}$, and multiplying this inequality by the nonnegative number $|x_{m}|_{V}$ (claim 5 of Elementary Arithmetic in an Ordered Field) gives $|x_{m}|_{V}\le|x_{m}|_{V}^{2}=2h(x_{m})\le2c\le1+2c$.

By Hilbert Triples: Standing Notation and Background (ii) the space $(V,d_{V})$ is separable, so Weak Compactness and Closedness of Bounded Subsets of the Small Space of a Hilbert Triple (ii), applied with $C=1+2c$, gives $x\in V$ and that $(x_{m})$ converges weakly to $x$ in $V$. By Weak Convergence of a Sequence in a Real Inner Product Space, the real sequence $(\langle x_{m},x\rangle_{V})_{m\in\mathbb{N}}$ converges to $\langle x,x\rangle_{V}=|x|_{V}^{2}$ (the last equality by Real Inner Product Spaces (ii)). For every $m$, Elementary Identities in a Real Inner Product Space (v) gives

$$0\le|x_{m}-x|_{V}^{2}=|x_{m}|_{V}^{2}-2\langle x_{m},x\rangle_{V}+|x|_{V}^{2},$$

the left inequality because $0\le|x_{m}-x|_{V}$ (Real Inner Product Spaces (ii)) and claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field, so that $2\langle x_{m},x\rangle_{V}\le|x_{m}|_{V}^{2}+|x|_{V}^{2}\le2c+|x|_{V}^{2}$. The left side converges to $2|x|_{V}^{2}$ by claim 3 of Arithmetic of Limits of Real Sequences, and the right side, a constant sequence, converges to $2c+|x|_{V}^{2}$ by Constant Sequences and Index-Shifted Sequences of Real Numbers (i), so claim 1 of Order Properties of Limits of Real Sequences gives $2|x|_{V}^{2}\le2c+|x|_{V}^{2}$, whence $|x|_{V}^{2}\le2c$ and $h(x)\le c$, that is, $t\le-h(x)$.

This verifies the condition of Functions with Closed Superlevel Sets: Sequential Characterisation, Semicontinuity, Perturbation and Limits (i), so $-h$ has closed superlevel sets in $H$. By A Real Function with Closed Superlevel Sets on a Subset of a Metric Space (ii) this means that for every $t\in\mathbb{R}$ the set $\{x\in V:t\le-h(x)\}=\{x\in V:h(x)\le-t\}$ is closed in $H$; as $t$ ranges over $\mathbb{R}$ so does $c=-t$, which is the first assertion. The sequential consequence stated in the claim is exactly what was proved above. Finally, Functions with Closed Superlevel Sets: Sequential Characterisation, Semicontinuity, Perturbation and Limits (ii) shows that $-h$ is upper semicontinuous on $V$, so $h=-(-h)$ is lower semicontinuous on $V$ by claim 1 of Semicontinuity Under Negation and Characterization of Continuity.

Claim 5. Let $x\in U$ and let $\varepsilon>0$. Since $U$ is open in $(H,d_{H})$, there is a real $\rho>0$ such that every $y\in H$ with $d_{H}(x,y)<\rho$ lies in $U$. Let $\eta$ be the lesser of $\varepsilon$ and $\rho$ (claim 9 of Elementary Order Arithmetic in an Ordered Field), so that $\eta>0$. By The Resolvent of the Form Operator of a Hilbert Triple: Minimisation, Contraction, and Density of the Domain (v) there is a real $\alpha_{0}>0$ with $|R_{\beta}x-x|_{H}<\eta$ for every real $\beta\ge\alpha_{0}$; taking $\beta=\alpha_{0}$, $|R_{\alpha_{0}}x-x|_{H}<\eta$, and by The Resolvent of the Form Operator of a Hilbert Triple: Minimisation, Contraction, and Density of the Domain (ii) the point $y=R_{\alpha_{0}}x$ lies in $D(A)$. Now $d_{H}(x,y)=|x-y|_{H}=|y-x|_{H}<\eta$ by Real Inner Product Spaces (iii) and Elementary Identities in a Real Inner Product Space (iv), so $y\in U$ because $\eta\le\rho$, and $|y-x|_{H}<\varepsilon$ because $\eta\le\varepsilon$. Thus $y\in D(A)\cap U$, and $y\in V\cap U$ as well since $D(A)\subseteq V$ (Hilbert Triples: Standing Notation and Background (iii)). If $U$ is nonempty, applying this to any point of $U$ shows that $D(A)\cap U$ and $V\cap U$ are nonempty.

Claim 6. By claim 5 the set $V\cap U$ is nonempty, so being bounded above or below near each point is meaningful for functions on the subset $V\cap U$ of $(H,d_{H})$, in the sense of Upper and Lower Semicontinuous Envelopes of a Real-Valued Function (i), which is the sense fixed by Real Hilbert Spaces: Standing Notation and Background (xvi). Suppose $u$ is bounded above near each point of $U$ and let $x\in V\cap U$. Since $x\in U$, there are $c\in\mathbb{R}$ and a real $r>0$ such that $u(y)\le c$ for every $y\in U$ with $d_{H}(y,x)\le r$. Let $y\in V\cap U$ satisfy $d_{H}(y,x)\le r$. Then $0\le\delta h(y)$ by claim 1 and $0<\delta$, so $u(y)-\delta h(y)\le u(y)\le c$ by claim 3 of Elementary Arithmetic in an Ordered Field and transitivity. Hence $c$ belongs to the set $A_{u-\delta h}(x)$ of Upper and Lower Semicontinuous Envelopes of a Real-Valued Function, formed for the function $u-\delta h$ on $V\cap U$ with the same radius $r$, which is therefore nonempty. As $x\in V\cap U$ was arbitrary, $u-\delta h$ is bounded above near each point of $V\cap U$. If instead $u$ is bounded below near each point of $U$, the same argument with $c\le u(y)$ for $y\in U$, $d_{H}(y,x)\le r$, gives $c\le u(y)\le u(y)+\delta h(y)$ for $y\in V\cap U$ with $d_{H}(y,x)\le r$, so $c\in B_{u+\delta h}(x)$ and $u+\delta h$ is bounded below near each point of $V\cap U$.