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Compact embedding and strong convergence

  1. Aug 4, 2010 #1
    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. jcsd
  3. Aug 6, 2010 #2
    Yes, it's due to the compactness of the embedding.
  4. Aug 21, 2010 #3
    Which assumption together with compactness imply this? Weak convergence in X or strong convergence in Y or both? Could you elaborate more?
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