If $Y\subset X$ are Banach spaces such that $Y$ is dense in $X$, is it true that $X'$ is dense in $Y'$?

With this $G$ is dense in $X^*$ in weak* sense if and only if $G$ is total set fact, the proof is rather easy.

Let me denote the embedding of $Y$ into $X$ by $i$. Then $i(Y)$ is dense in $X$. And the question is, whether $i^*(X')$ is weak-star dense in $Y'$.

Due to the result above, we have this density if and only if for all $y\in Y$ $$ (i^*f)(y) =0 \quad \forall f\in X' \Rightarrow y=0. $$ Now let $y\in Y$ be given such that $(i^*f)(y)=f(iy)=0$ for all $f\in X$. This implies $iy=0$, and by injectivity of $i$, $y=0$. So $i^*X'$ is total and hence dense in $Y'$.

Example. Sending each sequence to itself, we get $$ l^1 \longrightarrow c_0 $$ injective with dense range. Taking adjoint, we get $$ l^\infty \longleftarrow l^1 $$ injective but non-dense range.

Dense range of a bounded linear transformation by itself implies that the adjoint is injective.