Let $X$ and $Y$ be normed spaces and $T:X\to Y$ a bounded linear operator. The adjoint operator $T^\times:Y'\to X'$ of $T$ is defined by $$(T^\times g)(x)=g(Tx),\,g\in Y'$$ where $X'$ and $Y'$ are the dual spaces of $X$ and $Y$, respectively. I'm trying to prove that $\|T^\times\|=\|T\|$. I already got that $\|T^\times\|\leq\|T\|$, but I'm struggling to show that $\|T^\times\|\geq\|T\|$. In Kreyszig's Functional Analysis book, the author uses the following argument:
However, Theorem 4.3-3 (which resulted from the Hahn-Banach Theorem) states that:
If $X$ is a normed space and $x_0\neq 0$ is any element of $X$, then there exists a bounded linear functional $\tilde{f}$ on $X$ such that $$\|\tilde{f}\|=1\quad \text{and} \quad \tilde{f}(x_0)=\|x_0\|.$$
We know $x_0\neq 0$ does not imply that $T(x_0)\neq 0$, so how is this theorem applicable in this case?