Hey all I have this question: Let $X, Y$ be normed spaces, $T \in \mathrm{L}(X, Y)$. Prove the following. $T$ is compact if and only if $\overline{T\left(B_1(0)\right)}$ compact.
Proof:
Let $X, Y$ be normed spaces and $T \in \mathrm{L}(X, Y) $. We need to show that $ T $ is a compact operator if and only if $ \overline{T(B_1(0))} $ is compact in $ Y $.
First, assume that $ T $ is a compact operator. By the definition of compactness, $T $ maps bounded sets in $ X $ to relatively compact sets in $ Y $. The unit ball $ B_1(0) $ in $ X $ is a bounded set, so its image under $ T $, denoted by $ T(B_1(0)) $, is relatively compact in $ Y $. This means that the closure of $ T(B_1(0)) $, which is $ \overline{T(B_1(0))} $, is compact in $ Y $, as a set is relatively compact if and only if its closure is compact.
Conversely, suppose $ \overline{T(B_1(0))} $ is compact in $ Y $. To show that T is compact, we need to demonstrate that it maps every bounded set in ( X ) to a relatively compact set in Y . Consider any bounded set B in X. Since B is bounded, it can be enclosed within a scaled unit ball, say $ \lambda B_1(0) $ for some $ \lambda > 0 $. Since T is linear, we have $T(\lambda B_1(0)) = \lambda T(B_1(0)) $. Now, $ \overline{T(B_1(0))} $ is compact and compactness is preserved under scalar multiplication, so $ \overline{\lambda T(B_1(0))} $ is also compact. This implies that $ \lambda T(B_1(0)) $ is relatively compact because the closure of a relatively compact set is compact. Since $ T(B) \subseteq T(\lambda B_1(0)) $, and a subset of a relatively compact set is relatively compact, $ T(B) $ is relatively compact. Hence, T maps any bounded set in X to a relatively compact set in Y, establishing T as a compact operator.
Therefore, T is a compact operator if and only if $ \overline{T(B_1(0))} $ is compact in Y. This completes the proof, demonstrating the equivalence between the compactness of an operator and the compactness of the closure of its image of the unit ball in the domain space.
-- Can you please tell me if my working is alright? Thanks in advance :)