Suppose for each given natural number n I have a convergent sequence (y_i^{(n)}) (in a Banach space) which has a limit I'll call y_n and suppose the sequence (y_n) converges to y.
Can I construct a sequence using elements (so not the limits themselves) of the sequences (y_i^{(n)}) which...
Sorry for the rather vague title!
Homework Statement
Given:
Two Banach spaces A and B, and a linear map T: A\rightarrow B
The sequences (x^n_i) in A. For each fixed n, (x^n_i) \rightarrow 0 for i \rightarrow \infty.
The sequences (Tx^n_i) in B. For each fixed n, (Tx^n_i) \rightarrow y_n...