Isomorphism on Dual spaces by Bounded Linear Operator

Question

Let $X$ and $Y$ be Banach spaces and let $T:X→Y$ be an onto bounded linear operator. Show that $Y^*$ (the dual space of $Y$) is isomorphic to a subspace of $X^*$ (the dual space of $X$).

I am not sure which subspace of $X^*$ i should be looking at.

Please help. Your help is very much appreciated.

Answer 1

Let $T^*:Y^*\rightarrow X^*$defined by $T^*(y)(x)=\langle T(x),y\rangle$ show that $T$ is injective.