Then dual map of a linear operator.

Question

Let $X$ and $Y$ be normed spaces. Let $T:X\rightarrow Y$ be a linear map. Then the dual map $T^*:{Y}^*\rightarrow{X}^{*}$ is defined by $T^*(f)=f\circ T$. I want to show $T^*$ is well defined and it suffices to show $f\circ T\in {X}^{*}$. But I am stuck on how to show $f\circ T$ is bounded. Can anyone help me? Thank you!

Answer 1

If you don't assume $T$ to be bounded then in general $D(T^*) \neq Y^*$. You can define a dual operator by $T^*: D(T^*) \to X^*$ by setting $D(T^*) = \{y^* \in Y^* \colon \exists x^* \in X^* \text{ with } y^*(Tx) = x^*(x) \forall x \in D(T) \}$ and $T^*y^* = x^*$.