While studying weakly compact operators the book proves that every compact operator is weakly compact and also that every weakly compact operator is bounded, I got it, ad they conclude those results place weak compactness between compactness and boundedness as a property for linear operators between Banach spaces.
As counterexamples they say the identity operator in $l_1$ is a bounded operator ( I can see this) but isn't weakly compact (not so sure how to prove that) and that the identity operator in $l_2$ is weakly compact (why?) but isn't bounded.
The definition for a weakly compact operator is the following:
Let X,Y be Banach spaces, we say $T:X\longrightarrow Y$ is weakly compact, if for every bounded set $B\subset X$ , $T(B)$ is relatively weakly subset of $Y$.
Any guide in how to prove this? I don't want the full answer, thank you.