Weak convergence in $l_p$ implies pointwise convergence?

Question

Could someone please share their thoughts on this one:

Consider at $l_p(Y)$, for $1<p<\infty$ with the counting measure on $Y$.

Show that if a sequence weakly converges in $l_p(Y)$ then it would converge pointwise in Y. Show that the converse holds only when Y is finite.

Thanks!

Answer 1

Hint
Can you bound $|(x_n)_j - x_j| \le \|x_n - x\|_p$?
Take the canonical basis as a counter example for b). (what is it's point-wise limit? What is $\|i^p e_i - e\|_p$?)