I am working from Hilbert Space Methods for Partial Differential Equations by Showalter. Page 29 states, "Theorem 1.A shows that $T: L^1_{\text{loc}}(G) \to C_0^\infty(G)^*$ is an injection." Page 28 gives the cited theorem
Theorem 1.A $C_0^\infty(G)$ is dense in $L^p(G)$.
What I know: (1) The inclusion $i: C_0^\infty(G) \to L^p(G)$ induces dual map $i': L^p(G)' \to C^\infty_0(G)^*$. (2) $\text{Range}(i)$ dense in $L^p(G)$ implies that $i'$ is an injection. (3) For all $1 \leq p \leq \infty$, we have $L^p(G) \leq L^1_{\text{loc}}(G)$.
But I don't quite see how these facts fit together to give me the transformation $T: L^1_{\text{loc}}(G) \to C_0^\infty(G)^*$. Can $L^1_{\text{loc}}(G)$ perhaps be identified with a subspace of $L^p(G)'$?
Edit (in response to a comment)
The author also writes,
$f \in L^1_{\text{loc}}(G)$ is assigned the distribution $T_f \in C_0^\infty(G)^*$ defined by $T_f(\varphi) = \int_G f \overline{\varphi}$ for $\varphi \in C_0^\infty(G)$.
Somehow, it didn't click for me that this mapping $f \mapsto T_f$ is the mapping $T: L^1_{\text{loc}}(G) \to C_0^\infty(G)^*$. But still my question remains: What has density of $C_0^\infty(G)$ in $L^p(G)$ got to do with injectivity of $T$?
This is a rather a long comment that a precise proof of the Lemmas and Theorems involve. For sake of simplicity only real valued functions are considered.
In that textbook, the author uses $\mathcal{C}^\infty_0(G)$ for the space of smooth functions on $G$ that have compact support (some other books use $C^\infty_{00}(G)$ or $\mathcal{D}_G$. This is to clarify the meaning of things in your posting.
The Lemma quoted in valid for $0<p<\infty$ as an $F$-space for $0<p<1$ or as Banach space for $1\leq p<\infty)$. When $p=\infty$ the result holds but with $L_\infty$ is equipped with the weak topology $\sigma(L_\infty,L_1)$.
$\mathcal{D}_G$ however is typically equipped with a linear vector topology through an inductive limit of norms $$p_{m,K}(\phi):=\sup\{|\phi^{(\alpha)}(x)|:x\in K, 0\leq|\alpha|\leq m\}, \operatorname{supp}(\phi)\subset K$$ where $K\subset G$ is compact. (It is possible to write $G=\bigcup_nK_n$ with $\operatorname{int}(K_n)\subset K_{n+1}$. Let $\tau$ be that somewhat complicated topology.
Is in this context that the dual $\mathcal{D}'_G$ of $(\mathcal{C}^\infty_{00}(G),\tau)$ is defined.
The space of distributions $\mathcal{D}'_G$ contains many interstice functionals; amongst them them Radon measures (and this $L^{loc}_1(G)$.
To check that indeed that the inclusion map form $L^{loc}_1(G)$ into $\mathcal{D}'_G$ is injective, suppose $f,g\in L^{loc}_1(G)$ is such that $$\int_G f\phi=\int_G g\phi,\qquad \phi\in \mathcal{D}_G$$ Let $K\subset G$ a compact with nonempty interior and consider $\mathcal{D}_K$ (smooth functions with support in $K$). Appealing to the Lemma when $p=\infty$ (the domain being $K$ and the topology being the weak $\sigma(L_\infty,L_1)$ topology) or the Riesz representation theorem yields $$\int_K f\phi=\int_K g\phi,\qquad \phi\in L_\infty(K)$$ this by the Riesz representation theorem $f=g $ almost surely on any relatively compact subset $K$ of $G$, and thus on $G$.