If you notice the convergence is strong with respect to a functional: For example, in 4.4 it's asserted that for in a suitable space, and this is thedefinitionof weak convergence. The case of the other lemmas are analogous by Riesz-representation for or by deifnition of .

As for it's the space of strongly measurable functions such that .

I suggest you take a look at Evans' "Partial Differential Equations" before continuing.