Suppose that $\Vert \cdot \Vert$ is the norm induced by an inner product $\left \langle \cdot,\cdot \right \rangle$ defined on a vector space $X$. The question is, what is $\Vert \alpha \Vert$ equal to, where $\alpha$ is a scalar? The reason i ask is because I noticed my friend claimed that $\Vert \alpha \Vert^2=\left \langle \alpha,\alpha \right \rangle=\alpha\overline{\alpha}=\vert \alpha\vert^2$, and hence he $\Vert \alpha \Vert=\vert \alpha\vert$.
However, I disagreed, because we simply have $$ \Vert \alpha \Vert^2=\left \langle \alpha,\alpha \right \rangle, $$ and we can't what the right side is equal to, as we do not know what $\left \langle \cdot,\cdot \right \rangle$ is defined as, right? What I mean is that the result depends on the definition of $\left \langle \cdot,\cdot \right \rangle$.