I would like your help understanding this [article]. [article]: http://web.mit.edu/kjb/www/Principal_Publications/A_posteriori_Error_Estimation_Techniques_in_Practical_Finite_Element_Analysis.pdf

Page 239 (3.2 A priori error estimates), I am quickly getting lost because of the type of norm that is always changing.

Things I do not understand:

1. I never saw Cea's lemma written with an "inf", but I am guessing this is due to Galerkin's procedure.
2. For equation 14 and 15, I simply do not understand how they change the norms in the inequalities, and how they jumped from the H-norm result to the L2-norm result.

Thank you for your help !