Theorem IX.13 Let $f$ be in $\mathscr{L}^2(\mathbb{R}^n)$. Then $e^{b|x|}f \in \mathscr{L}^2(\mathbb{R^n})$ for all $b<a$ if and only if $\hat f$ has an analytic continuation to the set $\{ \zeta \mid |\mathrm{Im}{\zeta}| < a \}$ with the property that for each $\eta\in\mathbb{R}^n$ with $|\eta| < a$, $\hat{f}(\cdot + i\eta)\in\mathscr{L}^2(\mathbb{R}^n)$ and for any $b < a$, $$\sup_{|\eta|\leq b} \Vert \hat{f}(\cdot + i\eta) \Vert_{2} < \infty.$$
I'm failing to see why it is required to make the additional assumption that $$\sup_{|\eta|\leq b} \Vert \hat{f}(\cdot + i\eta) \Vert_{2} < \infty.$$ Does the fact that $\hat{f}(\cdot + i\eta)\in\mathscr{L}^2(\mathbb{R}^n)$ for $|\eta| < a$ not imply it automatically? Or am I missing something?