Compact embedding and strong convergence

[NSAC]

Let X be compactly embedded in Y. Assume also that
there is a sequence f_n in X such that
f_n converges to f weakly in X and strongly in Y to some function f in X.
Can we say that f_n converges to f strongly in X?

 

[Eynstone]

Yes, it's due to the compactness of the embedding.

 

[NSAC]

Yes, it's due to the compactness of the embedding.

Which assumption together with compactness imply this? Weak convergence in X or strong convergence in Y or both? Could you elaborate more?