weakly convergence imply strong convergence when $ \|f_n\| \rightarrow \|f\| $ in $l^2([0,1])$?

Question

I know in general weakly convergence do not imply strong convergence in $L^p$,but in $L^2[0,1]$ space which if we have additional condition do this condition plus the weak convergence will give us strong convergence?

The additional condition is $f \in L^2[0,1]$ and $ \|f_n\| \rightarrow \|f\| $

Answer 1

This is true in an arbitrary Hilbert space:

$$\|f_n - f\|^2 = \langle f_n-f,f_n-f\rangle = \langle f_n, f_n\rangle -\langle f_n,f\rangle - \langle f,f_n\rangle +\langle f,f\rangle\to 0.$$