If $Ω$ is a bounded domain and $K⊆Ω$ is compact, how can we define the restriction of a continuous linear functional $C^0(\overlineΩ)→ℝ$ to $C^0(K)$?

Question

Let $d\in\mathbb N$, $\Omega\subseteq\mathbb R^d$ be a bounded domain and $K\subseteq\Omega$ be compact. In Remark B.23 of Finite Element Methods for Incompressible Flow Problems the author is writing about the restriction of a continuous linear $\Phi:C^0(\overline\Omega)\to\mathbb R$ to $C^0(K)$.

How is such a restriction defined?

It's clear that, by Tietze's extension theorem, any function from $C^0(K)$ can be extended to $C^0(\overline\Omega)$. However, such an extension is not unique and hence I have no idea how $\Phi$ can be restricted to $C^0(K)$.

Let me note that, in the context of the book, $\Phi(p)$ might be the evaluation of $p\in C^0(\overline\Omega)$ at some point $x\in\overline\Omega$. So, if $x\not\in K$, then the nonuniqueness of the formerly mentioned extensions should be a problem.

Answer 1

One possibility is to use the Riesz Representation Theorem.

Theorem: Suppose $X$ is locally-compact. Every linear functional on $C^0(X)$ is of the form $\displaystyle f \mapsto \int _Xf(x) d \mu (x)$ for some (signed) Borel measure $\mu$ on $X$.

So given a functional $\Phi$ on $C^0(\overline \Omega)$ we get a measure $\mu$ on $\overline \Omega$. We then define the measure $\nu$ on $K$ to be the restriction of $\mu$ to $K$. This allows us to define the functional $\Psi$ given by $\displaystyle g \mapsto \int _K g(y) d \nu (y)$ on $C^0(K)$.

In case $\Psi$ is evaluation at some point $x \in \overline \Omega - K$ the corresponding measure is the point-measure at $x$. The restriction of that measure to $K$ is the zero measure and we get $\Psi \colon C^0(K) \to \mathbb R$ is the zero functional.

Edit: It feels like $\Psi$ should be obtainable from $\Phi$ in the following manner: Given a continuous function $f$ on $K$ define the function $g$ on $\overline \Omega$ to coincide with $f$ over $K$ and to be zero elsewhere. We cannot just define $\Psi(f) = \Phi(g)$ because $g$ is most likely not continuous. So instead choose any series of continuous functions $g_n$ on $\overline \Omega$ that converges pointwise to $g$. Define $\Psi (f) = \lim \Phi(g_n)$

Answer 2

The space $C^0(K)$ is commonly defined as the collection of all $f \in C(\mathbb R^d)$ whose support is a subset of $K$. If $K \subset \Omega$ then you have $$C^0(K) \subset C^0(\overline \Omega)$$ and the term "restriction" has an unambiguous meaning.