How can "$L_2(\Omega) \subset D'(\Omega)$" not be an aberration in term of logical symbolism ? $*_1$
I get that , the idea is to identify a L_2 function to it's regular associated distribution and to continusously inject the latter into D' , but i'm also very "shocked" that the logical symbol of "$\subset$" has been defined (in most books i've read) by : $B \subset A \Leftrightarrow(\forall x)(x \in B \rightarrow x\in A)$ I'd see something like , maybe it exists a terminology for the symbol "$\subset$" such as , for two sets of differents objects , it is defined by "it exists a continuous injection from the first set to the second";
$*_1$: D'($\Omega$) is a set of function of $D(\Omega)\rightarrow R$ ,$D(\Omega)$ being itself a set of function ,$L_2$($\Omega$) is a set of function of $\Omega\rightarrow R$ , so it defines two different mathematical objects in my opinion.
Edit : $L_2$($\Omega$) is a set of Class of functions,
Do you think that $\mathbb{Q} \subseteq \mathbb{R}$? Technically, these are completely different beasts: $\mathbb{Q}$ consists of pairs of integers (the second component non-zero) subject to a certain equivalence relation. $\mathbb{R}$ consists (depending on your taste) Cauchy sequences subject to a certain equivalence relation or Dedekind cuts (additional possibilities exist). These are certainly different types of mathematical objects but I think most people would agree that having $\mathbb{Q} \subseteq \mathbb{R}$ is useful. For example, we can now take the square root of a (positive) rational number even though it might not be rational again.
So here’s what we do: We find a copy of $\mathbb{Q}$ within $\mathbb{R}$ (i.e. an injection $\mathbb{Q} \hookrightarrow \mathbb{R}$) and then essentially redefine $\mathbb{Q}$ to mean the image of this injection. This is what we mean when we say that we “identify” the two sets. (We keep the original definition around because it often makes thinking about these objects easier.) One thing that is important here is that the inclusion is canonical, so there is no confusion which real number I mean when I talk about the rational number $\frac{23}{42}$.
The inclusion $L_2(\Omega) \subseteq D'(\Omega)$ works the same way. Here the canonical inclusion is given by the regular associated distribution you mentioned. But sometimes it is more convenient to just think of $L_2$-functions as distributions so you can apply stuff to it that generally only works for distributions (like the square root above).
This is a powerful tool but one also needs to be a bit careful with this trick. You shouldn’t simply apply it to any injection you see. For example, if you have three sets $A$, $B$, $C$ and injections $i : A \hookrightarrow B$, $j : B \hookrightarrow C$ and $k : A \hookrightarrow C$, the composition $j \circ i$ need not agree with $k$. If you use all three to make identifications $A \subseteq B \subseteq C$ and $A \subseteq C$ you can now consider elements $a \in A$ as elements of $C$ in two different ways; if these don’t agree, things get confusing.