Let $X$ be a Normed Linear Space. The dual space $X^*$ of $X$ is the set of all bounded linear functionals on $X$. It is a normed linear space with the norm $\Vert\varphi\Vert$. (according to notes).
I can show the triangle inequality is true.
Trying to show $\Vert\varphi\Vert = 0$ iff $\varphi = 0$, I get:
$\varphi = 0$ implies $\Vert\varphi\Vert = 0$. (OK).
$\Vert\varphi\Vert = 0$ implies $\varphi (x) = 0$ for $\Vert\varphi\Vert\leq 1$. But I don't see how this implies $\varphi = 0$.
Also, $\Vert\lambda\varphi(x)\Vert = \vert \lambda \vert \sup\{\vert\varphi(x)\vert \colon \vert\lambda \vert \Vert x \Vert\leq 1\} \neq \vert\lambda\vert \Vert\varphi(x)\Vert$
Where have I gone wrong?