Proof of The $\delta$-Envelopes on an Open Subset, under Penalisation of a Continuous Function, and on a Closed Subset

Each result cited below is universally quantified over the data appearing in its own statement, and is applied here to the data named at the point of use. Since $U$ is nonempty and open in $H$, the sets $D(A)\cap U$ and $V\cap U$ are nonempty by The Penalty Function $h=\tfrac12|\cdot|_V^2$ of a Hilbert Triple: Expansion Identities, Closed Sublevel Sets, Density and Local Bounds (v), and the same applies to any nonempty $U'\subseteq U$ open in $H$; so all the $\delta$-envelopes named below are defined once the stated local bounds hold.

We record two remarks used repeatedly.

Remark A (restriction). Let $B'\subseteq B\subseteq H$ with $x\in B'$, and let $\vartheta:B\to\mathbb{R}$. If $\vartheta$ is continuous, upper semicontinuous or lower semicontinuous at $x$ relative to $B$, then the restriction of $\vartheta$ to $B'$ has the same property at $x$ relative to $B'$. Indeed, each of those three conditions asserts the existence, for a given positive $\varepsilon$, of a positive $\delta$ such that an inequality holds for every point $y$ of the set relative to which the property is asserted with $d_{H}(x,y)<\delta$; since $B'\subseteq B$, the same $\delta$ serves.

Remark B (ambient space). Each of those three conditions, and likewise convergence of a sequence, constrains only values of the metric at pairs of points lying in the set in question. Consequently, for $A\subseteq X\subseteq H$, a function on $A$ has one of those properties at a point relative to $A$ in the metric space $(H,d_{H})$ if and only if it has it in the metric space $(X,d_{H})$, the restriction of $d_{H}$ to $X$ being a metric by claim 1 of The Restriction of a Metric to a Subset Induces the Subspace Topology; and a sequence in $A$ converges to a point $x\in X$ in $(X,d_{H})$ if and only if it converges to $x$ in $(H,d_{H})$.

Claim 1. Let $U'\subseteq U$ be nonempty and open in $H$, and suppose $u$ is bounded above near each point of $U$. We apply Local Bounds, Semicontinuous Envelopes and Local Extrema Are Unchanged on the Trace of an Open Set twice, in both cases with the metric space $(H,d_{H})$ and the open set $O=U'$.

First, with $S=U$: here $S\cap O=U\cap U'=U'$, and claim 1 of that lemma shows that $u|_{U'}$ is bounded above near each point of $U'$.

Secondly, with $S=V\cap U$ and the function $V\cap U\to\mathbb{R}$ whose value at $x$ is $u(x)-\delta h(x)$, which is bounded above near each point of $V\cap U$ by The Penalty Function $h=\tfrac12|\cdot|_V^2$ of a Hilbert Triple: Expansion Identities, Closed Sublevel Sets, Density and Local Bounds (vi): here $S\cap O=(V\cap U)\cap U'=V\cap U'$, since $U'\subseteq U$. Claim 2 of that lemma gives, for every $x\in V\cap U'$,

$$\bigl((u-\delta h)|_{V\cap U'}\bigr)^{*}(x)=(u-\delta h)^{*}(x),$$

where $u-\delta h$ abbreviates the function just named. The right-hand side is $u^{-}_{\delta}(x)$ by The $\delta$-Envelopes $u^-_\delta$ and $u^+_\delta$ of a Function on an Open Subset of a Hilbert Triple (i). The left-hand side is $\bigl(u|_{U'}\bigr)^{-}_{\delta}(x)$ by the same reference applied to the function $u|_{U'}$ on $U'$: the function it attaches to $u|_{U'}$ is the function on $V\cap U'$ with value $u(x)-\delta h(x)$ at $x$, which is precisely $(u-\delta h)|_{V\cap U'}$. This proves the first half of the claim; the second half is obtained in the same way, using the second half of The Penalty Function $h=\tfrac12|\cdot|_V^2$ of a Hilbert Triple: Expansion Identities, Closed Sublevel Sets, Density and Local Bounds (vi), claim 3 of Local Bounds, Semicontinuous Envelopes and Local Extrema Are Unchanged on the Trace of an Open Set and The $\delta$-Envelopes $u^-_\delta$ and $u^+_\delta$ of a Function on an Open Subset of a Hilbert Triple (ii).

Claim 2. Let $\psi:U\to\mathbb{R}$ be continuous on $U$ and let $0<\lambda$, so that $0\le\lambda$.

Local bounds for $\psi$. Let $x\in U$. Applying Continuous Map Between Metric Spaces with $\varepsilon=1$, there is a positive $\delta_{0}\in\mathbb{R}$ such that every $y\in U$ with $d_{H}(x,y)<\delta_{0}$ satisfies $|\psi(y)-\psi(x)|<1$, hence $\psi(y)<\psi(x)+1$ and $\psi(x)-1<\psi(y)$ by claim 9 of Properties of the Absolute Value in an Ordered Field. Put $r=\tfrac{\delta_{0}}{2}$, which is positive and satisfies $r<\delta_{0}$ by claim 8 of Elementary Order Arithmetic in an Ordered Field. Every $y\in U$ with $d_{H}(y,x)\le r$ satisfies $d_{H}(x,y)=d_{H}(y,x)\le r<\delta_{0}$ by the symmetry axiom and claim 2 of Elementary Order Arithmetic in an Ordered Field, hence $\psi(y)\le\psi(x)+1$ and $\psi(x)-1\le\psi(y)$. So $\psi$ is bounded above and bounded below near each point of $U$, by Upper and Lower Semicontinuous Envelopes of a Real-Valued Function (i).

Upper semicontinuity. By The Penalty Function $h=\tfrac12|\cdot|_V^2$ of a Hilbert Triple: Expansion Identities, Closed Sublevel Sets, Density and Local Bounds (iv), $h$ is lower semicontinuous on $V$; by Remark A its restriction to $V\cap U$ is lower semicontinuous on $V\cap U$, so by claim 1 of Semicontinuity Under Negation and Characterization of Continuity the function $V\cap U\to\mathbb{R}$ with value $-h(x)$ at $x$ is upper semicontinuous on $V\cap U$, and by claim 2 of Sums and Nonnegative Multiples of Semicontinuous Functions so is its multiple by $\lambda$, whose value at $x$ is $-\lambda h(x)$. By Remark A and claim 2 of Semicontinuity Under Negation and Characterization of Continuity, the restriction of $\psi$ to $V\cap U$ is upper semicontinuous on $V\cap U$. By claim 1 of Sums and Nonnegative Multiples of Semicontinuous Functions their sum, the function with value $\psi(x)-\lambda h(x)$ at $x$, is upper semicontinuous on $V\cap U$.

Local bounds and the envelope. Since $\psi$ is bounded above near each point of $U$, The Penalty Function $h=\tfrac12|\cdot|_V^2$ of a Hilbert Triple: Expansion Identities, Closed Sublevel Sets, Density and Local Bounds (vi), applied to $\psi$ with $\lambda$ in the role of its $\delta$, shows that the function with value $\psi(x)-\lambda h(x)$ at $x$ is bounded above near each point of $V\cap U$; its upper semicontinuous envelope is therefore defined, and equals the function itself by claim 4 of Properties of the Upper Semicontinuous Envelope.

The statement for $\psi+\lambda h$ is obtained in the same way: $h$ restricted to $V\cap U$ is lower semicontinuous there, hence so is its multiple by $\lambda$ and its sum with the restriction of $\psi$, by claim 3 of Sums and Nonnegative Multiples of Semicontinuous Functions used twice together with claim 2 of Semicontinuity Under Negation and Characterization of Continuity; the function is bounded below near each point of $V\cap U$ by the second half of The Penalty Function $h=\tfrac12|\cdot|_V^2$ of a Hilbert Triple: Expansion Identities, Closed Sublevel Sets, Density and Local Bounds (vi); and it equals its own lower semicontinuous envelope by claim 5 of Properties of the Lower Semicontinuous Envelope, by Duality.

Claim 3. Suppose $u$ is bounded above near each point of $U$, let $K\subseteq U$ be closed in $H$ and let $\vartheta:U\to\mathbb{R}$ be continuous on $U$. Write $\Phi$ for the function $V\cap K\to\mathbb{R}$ with value $u^{-}_{\delta}(x)-\vartheta(x)$ at $x$.

By Basic Properties of the $\delta$-Envelopes: Semicontinuity, Duality, Closed Superlevel Sets, Bounds and Monotonicity (iii), $u^{-}_{\delta}$, as a function on the subset $V\cap U$ of the metric space $(U,d_{H})$, has closed superlevel sets in $(U,d_{H})$. By Remark B, $\vartheta$ is continuous on $U$ as a map from the metric space $(U,d_{H})$ to $(\mathbb{R},d_{\mathbb{R}})$. Claim 3 of Functions with Closed Superlevel Sets: Sequential Characterisation, Semicontinuity, Perturbation and Limits, applied in the metric space $(U,d_{H})$ with $A=V\cap U$, therefore shows that the function $V\cap U\to\mathbb{R}$ with value $u^{-}_{\delta}(x)-\vartheta(x)$ at $x$ has closed superlevel sets in $(U,d_{H})$.

Now let $t\in\mathbb{R}$, let $(x_{m})_{m\in\mathbb{N}}$ be a sequence in $V\cap K$ converging to a point $x\in H$ in $(H,d_{H})$, and suppose $t\le\Phi(x_{m})$ for every $m$. Since $K$ is closed in $H$ and $x_{m}\in K$ for every $m$, Sequential Characterization of Closed Subsets of a Metric Space gives $x\in K$, hence $x\in U$. By Remark B the sequence converges to $x$ in $(U,d_{H})$, and it lies in $V\cap U$ because $K\subseteq U$. Claim 1 of Functions with Closed Superlevel Sets: Sequential Characterisation, Semicontinuity, Perturbation and Limits, applied in $(U,d_{H})$ to the function of the previous paragraph, therefore gives $x\in V\cap U$ and $t\le u^{-}_{\delta}(x)-\vartheta(x)$; since also $x\in K$, we have $x\in V\cap K$ and $t\le\Phi(x)$. Claim 1 of Functions with Closed Superlevel Sets: Sequential Characterisation, Semicontinuity, Perturbation and Limits, applied now in the metric space $(H,d_{H})$ with $A=V\cap K$, shows that $\Phi$ has closed superlevel sets in $H$.

For the second half, suppose $u$ is bounded below near each point of $U$. By the second half of Basic Properties of the $\delta$-Envelopes: Semicontinuity, Duality, Closed Superlevel Sets, Bounds and Monotonicity (iii), the function $V\cap U\to\mathbb{R}$ with value $-u^{+}_{\delta}(x)$ at $x$ has closed superlevel sets in $(U,d_{H})$. The function $-\vartheta$ with value $-\vartheta(x)$ at $x$ is continuous on $U$, since $|(-\vartheta)(y)-(-\vartheta)(x)|=|\vartheta(y)-\vartheta(x)|$ by claim 2 of Properties of the Absolute Value in an Ordered Field, so Continuous Map Between Metric Spaces is satisfied with the same $\delta$ as for $\vartheta$. Since $\vartheta(x)-u^{+}_{\delta}(x)=(-u^{+}_{\delta}(x))-(-\vartheta(x))$, the argument above applies verbatim with these two functions in place of $u^{-}_{\delta}$ and $\vartheta$.