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.

when $T$ is a linear operator $X \to Y$ between two normed vector spaces 
$$\max_{\|z\| \le 1} \|T z\| \qquad\qquad \text{and} \qquad\qquad\max_{\|z\| = 1} \| T z \|$$  are the same. 
this is because a norm asks for $\| a z\| = |a| \ \|z\| $ for any constant $a$ and $T$ is linear so  $\| T (a z)\| = |a| \ \|T z\| $.
here your linear operator is $$\begin{array}{ll}T : &X \to \mathbb{R} \ \ \ (\text{ or } \mathbb{C}) \\&z \to \langle w,z\rangle\end{array}$$ and its (operator) norm is $$\| T\| = \max_{\|z\|=1} | T z| = \max_{\|z\|=1} | \langle w,z\rangle | $$
here it is not the case but don't forget many linear operators are unbounded, sometimes  $\|T\| = \infty$ 
