I was reading Compact Operator from the functional analysis Book by Erwin Kryszig. There I find that if $x_n\rightharpoonup x_0$ in a Banach space $X$, then for a compact operator $T$, $Tx_n \to Tx_0$. An important portion of the proof consists of the fact that $Tx_n \rightharpoonup Tx_0$.
Now, my question is, can we prove the same assertion, without using the fact that $Tx_n \rightharpoonup Tx_0$?
A detailed answer will be of very much help. Thanks in advance.