Condition on definition of dual norm In a review I'm reading, the dual norm is defined as $$||z||_*=\max_{||w||\leq 1}\langle w,z\rangle.  $$
Though I'm having a hard time understanding why $||w||\leq 1$ isn't equivalent to $||w||=1$, as given a $w$ such that $\langle w,z\rangle\geq 0$, if $||w||<1$, then using $\tilde{w}=1/||w||*w$ will give a larger inner product.

\begin{equation}
\begin{aligned}
\parallel u \parallel_{*} & = \enspace \max\limits_{\parallel x \parallel \leq 1} <u, x>\\
\leq & \enspace \max\limits_{\parallel x \parallel \leq 1} \frac{<u, x>}{\parallel x \parallel}\\
\leq & \enspace \max\limits_{\parallel x \parallel \neq 0} \frac{<u, x>}{\parallel x \parallel}\\
= & \enspace \max\limits_{\parallel y \parallel = 1} <u, y>\\
\leq & \enspace \max\limits_{\parallel y \parallel \leq 1} <u, y> = \enspace \parallel u \parallel_{*}
\end{aligned}
\end{equation}
Essentially, this means 
$$
\parallel u \parallel_{*} = \max\limits_{\parallel x \parallel \neq 0} \frac{<u, x>}{\parallel x \parallel}
$$
and it implies that norm of "other vector" is exactly 1. I am guessing there is some corner case (may be depending on space, norm...) for which the inequality is needed. But for the normal cases, we can just assume that we are dealing with unit vector to find dual norm of $u$.
