Do weak convergence and convergence of norms imply convergence in $L^2$?
It's almost surely a duplicate, but I think answering is shorter than finding the corresponding one.
Hint: we have $\lVert f-f_n\rVert_{L^2}^2=\lVert f\rVert_{L^2}^2-2\langle f,f_n\rangle+\lVert f_n\rVert_{L^2}^2$. The second term converges to $2\lVert f\rVert_{L^2}^2$.
In every Uniformly Convex Banach space this is true. See for example the book of Brezis Proposition 3.32 page 78: Brezis, Haim Functional analysis, Sobolev spaces and partial differential equations. Universitext. Springer, New York, 2011. xiv+599 pp