Show that every closed linear subspace $ \ Y \ $ of a Banach space $ \ X \ $ is weakly sequentially closed ,
that is , $ \ Y \ $ contains weak limits of all weakly convergent sequences $ \ \{y_n \} \ $ in $ \ Y \ $.
Answer:
We know closed subspace of a Banach space is also a Banach space.
Thus $ \ Y \ $ is a Banach space or Completely normed linear space.
That means every cauchy sequence in $ \ Y \ $ converges in it.
Now let $ \ \ y_n \to \ y \ $ weakly.
To finish the proof , we have to show that the weak limit $ \ y \ \in Y $ .
But I got no way to show it.
Help me doing this.