Range of double adjoint of a compact operator

Question

Let $X, Y$ be Banach spaces and $X^*$, the dual of $X$, has the approximation property. $T:X\rightarrow Y$ is a compact operator.
How can I show that the range of $T^{**}$, double adjoint of $T$, belongs to $Y$ without approximate $T$ by a sequence of finite rank operators?(Because I wanna show $T$ is approximated by these operators using the range of $T^{**}$)

Answer 1

A quick way to prove this is to use the celebrated theorem of Davis, Figiel, Johnson, and Pelczynski that weakly compact operators factorize over reflexive Banach spaces.

An easier proof: $A=\overline{T(B_X)}$ is compact in $Y$ and hence $J_Y(A)$ is $\sigma(Y^{\ast\ast},Y^*)$-closed in $Y^{\ast\ast}$ (where $J_Y:Y\to Y^{\ast\ast}$ is the canonical embedding). On the other hand, $B_X$ is $\sigma(X^{\ast\ast},X^*)$-dense in $B_{X^{\ast\ast}}$ and the continuity of $T^{\ast\ast}$ implies $T^{\ast\ast}(B_{X^{\ast\ast}})\subseteq A$ so that range$(T^{\ast\ast}) \subseteq J(Y)$.

Both proofs only need that $T$ is weakly compact.