Let $ 1 \leq p < \infty $.
Consider the normed space $\ell^p$ . Show that the following map is continuous. $$T(x_1,x_2,\ldots,x_n,\ldots) = (x_1^2,x_2^2,\ldots,x_n^2,\ldots) $$
Now let $p=1$. Consider $$A = \{x \in \ell^1 : |x_k| \leq \frac 1k \; \forall k \}$$
Determine if $T(A)$ is compact.
This is a question I found in a book and so it's missing a few details. I'm assuming here that $T$ is defined as map from $\ell^p$ to itself.
I tryed proving continuity by somehow going after Lipschitz but I didnt get very far.
As for the set $A$ , I know that the set is not compact in $\ell^1$ . I'm not sure how to determine if the Image is compact though.
Thoughts?
Hint for convergence: $$\|T(x) - T(y)\|^p = \sum_j |x_j - y_j|^p |x_j + y_j|^p \le \|x - y\|^p \sup_j |x_j + y_j|^p$$
Hint for $T(A)$: a sequence in $T(A)$ has a subsequence that converges elementwise. Does it converge in norm?