Compact embedding and strong convergence

1. Aug 4, 2010

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?

2. Aug 6, 2010

Eynstone

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

3. Aug 21, 2010

NSAC

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