I am currently studying distribution theory, with an eye towards PDEs, and have a question on regarding locally integrable functions as distributions.
Suppose $u\in L^1_{\mathrm{loc}}(\Omega).$ Then we can map $u$ to the space of distributions $D'(\Omega)$ by $$T(u)=I_u\in D'(\Omega),$$ where the distribution $I_u$ is defined by $$\langle I_u,\phi\rangle:=\int_{\mathbb{R}^n}\!f(x)\phi(x)\ \mathrm{d}x.$$ Given this definition, it is technically not correct to say $u\in D'(\Omega).$ Nonetheless, many textbooks consider $u$ to be a distribution. Salsa (PDEs in Action, pp. 375) even goes as far as to say $L^1_{\mathrm{loc}}(\Omega)\hookrightarrow D'(\Omega)$, which seems to assume $L^1_{\mathrm{loc}}(\Omega)\subset D'(\Omega)$. Furthermore, this would imply $$|| u||_{D'(\Omega)}\le C ||u ||_{L^1_{\mathrm{loc}}(\Omega)},$$ but what meaning can be assigned the the left hand side of the above equation if $u\not\in D'(\Omega)$?
As far as I can tell, when one writes $L^1_{\mathrm{loc}}(\Omega)\hookrightarrow D'(\Omega)$, to mean that the mapping $T:L^{1}_{\mathrm{loc}}(\Omega)\to D'(\Omega)$ is injective, and $\mathrm{im}(T)\hookrightarrow D'(\Omega)$?
The only thing I can think of is that this is just an abuse of notation, tolerated so one can write $$\Delta u=f,$$ and understand $u$ as a distribution. This would be more comfortable and convenient than writing $$\Delta I_u=f.$$
Am I thinking along the right lines? Am I crazy? I have done some searching, but most sources just seem to brush this off by saying they "regard" (?) $u$ as a distribution. I would very much appreciate any help offered.