# Is it true that $P = coNP$?

I have recently read the definition of $$coNP := \{L \ \mid \ \overline{L} \in P\}$$, where $$L$$ denotes a language. However, I am wondering what the difference between $$P$$ and $$coNP$$ is, since an elementary result of the theory of Turing Machines is that if $$L \in P$$, then $$\overline{L} \in P$$.

So is $$P = coNP$$ or is there something that I am missing here?

$$P=coNP$$ is an open question, equivalent to the famous: "is $$P=NP$$ true?" question.
The definition of $$coNP$$ you have is incorrect. The correct definition is the following:
$$coNP:=\{L\mid \overline{L}\in NP\}$$ As you can see, it reads $$NP$$ and not $$P$$. It was probably a typo wherever you saw that definition.