I have found many resources talking about the $l_2$ convergence of OLS estimation error and prediction error.
I have also read about the analysis of the $l_2$ convergence of lasso estimation error and prediction error, variable selection consistency, and $l_{\infty}$ convergence of lasso estimation error. Suppose I'm dealing with linear functions, then I can easily use the $l_{\infty}$ convergence of lasso estimation error to get a $l_{\infty}$ convergence of lasso prediction error, which must has the term $||X||_{\infty}$ involved.
My goal is to study the analysis of the $l_{\infty}$ convergence of OLS prediction error for more general functions. However I didn't find any relevant resources yet. To that end, I was hoping to find some textbook discuss the $l_{\infty}$ convergence on linear functions. But all I can find is the $l_2$ convergence of the prediction error. Then I think maybe I can just target on studying the $l_{\infty}$ convergence of estimation error, and then use the device $||X(\hat{\theta} - \theta^*)||_{\infty} \le ||X||_{\infty} ||\hat{\theta} - \theta^*||_{\infty}$.
However, I couldn't find any resources talking about the $l_{\infty}$ convergence of OLS estimation error. May I ask for any advice?
I understand the reason why the majority of textbooks focus on teaching the $l_2$ convergence - because the underlying objective is the means square error. Therefore, it makes sense to measure how close the mean response $X \hat{\theta}$ to the true parameter $X\theta^*$ in terms of $l_2$ error. But my goal is to understand the upper bound of the entries of $X(\hat{\theta} - \theta^*)$.