The term "as close as possible" is quantified by the error treshold I mentioned in the answer. A definition of such quantity can be found on page 2, eq. (2), on this pre-print (arxiv.org/pdf/2108.06558.pdf). I did cite myself because it was the quickest reference I could think of but there is a ton of further literature out there that is perhaps more explanatory than that.

The first infimum is taken over all the subspaces (of dimension n) of the solution manifold as you correctly pointed out. Then you take the sup of all the solution in the FOM so that you consider "the worst case scenario" as in of all the solutions in the FOM consider that with the highest energy (in terms of dynamical systems) and then subrtract it pointwise (i.e. for each entry in your vector of function values) from the infimum of low-rank solutions. The output will thus be the minimum number (first inf) of low-rank dimensions n that makes g "as close as possible" to f.

Part 2 If we instead specify P2-elements, then two neighbouring elements must agree on the nodal values of both vertices and the edge's midpoint. So when you say "If you now have shape functions on neighboring cells that agree in their value at the nodes of the common edge between the two cells..." it clearly holds for higher-order Lagrangian elements. Nonetheless what I don't understand is how the choice of the local basis functions enforce this condition that you mentioned.

Thanks @Wolfgang . Yes I got that the edge restriction of the elemental linear basis functions are themselves linear and thus they are uniquely identified by their nodal values; it is also clear to me that to glue the solution on two neighbouring elements they must agree on those nodal values for the edge that they share. However, as stated in my post, my doubt concerns how and when this vey constraint is enforced in the first place. How the choice of local basis functions shaped like that translates in having two elements to agree on the nodal values of the shared edge? Continue in Part 2

Thanks @WolfgangBangerth; do you mean the text doesn't make sense the way it's written?