One of the definitions of compact operators is:
For any bounded sequence $(x_{n})_{n\in \mathbb {N} } \in X$, the sequence $(Tx_{n})_{n\in \mathbb {N} }$ contains a converging subsequence.
Does this mean that the limit is in $Img(T)$? My problem is understanding exactly where the limit, which exists, resides, especially in light of the equivalent definition that $\overline{T(X_1)}$ is compact (where $X_1$ are all $x$ such that $||x||\leq 1$) - is the limit perhaps in $\overline{T(X_1)}$ or $T(X_1)$?
