Let $X$ and $Y$ be normed spaces, I've read that if $T:X^*\to Y$ is a bounded linear operator such that $T^*(Y^*)\subset X$ then $T$ is weak*-to-weak continuous.
First question. Does $T^*(Y^*)\subset X$ mean to be $T^*(Y^*)\subset J_X(X)$, where $J_X$ is the canonical embedding from $X$ to $X^{**}$?
Second question. Why should be such an operator weak*-to-weak continuous?
If $x_i^{*} \to x^{*}$ in weak* topology of $X^{*}$ then $(Tx_i^{*}) (y^{*})=x_i^{*}(T^{*}(y^{*})) \to x^{*}(T^{*}(y^{*}))=(T^{*}x^{*})(y^{*})$ for all $y^{*}$, so $Tx_i^{*} \to Tx^{*}$ in weak topology.
Since these topologies are not metrizable you have to use nets rather than sequences.
[$x_i^{*}(T^{*}(y^{*})) \to x^{*}(T^{*}(y^{*}))$ follows from the hypothesis that $T^{*}(Y^{*}) \subseteq X$].