I want to get into the habit of correctly using rigorous notation where possible. I have tried to do this as best as possible with the maximum likelihood estimator, but I am having some slight confusion as a lot of books will reference $\hat{l}(\theta;x_1,...,x_n)$.
I have been stating the following when talking about the MLE of variable $\theta$:
$\{\hat{\theta}_{MLE} \} \subseteq \{\text{argmax}_{\theta \in\Theta}L(\theta;x_1,...x_n)\}$
Where $L(\theta;x_1,...x_n)$ is the Likelihood function, $x_1,...x_n$ are the sample of iid observations from the distribution, and $\Theta$ refers to the whole parameter space.
Does this look adequate, to me it makes sense, but again I don't know if my notation is off. Any confirmation would be appreciated, thanks!