In the proof of the statement that weak convergence is equivalent to strong convergence in finite dimensional normed vector space, we need to use the fact that in finite dimensional space convergence in coordinates is equivalent to convergence in norm.
Since in infinite dimensional normed vector space weak convergence is generally different from strong convergence, I guess in infinite dimensional case convergence in coordinates is NOT equivalent to convergence in norm. But I have no example. Could someone give me an example of “convergence in coordinates is not equivalent to convergence in norm”?
Thanks so much.
Update: I think the infinite dimensional space must be separable, for otherwise it does not make sense to talk about “convergence in coordinates”