I need to show $\sum|\hat{f}(n)|< \infty$ if $f$ is continuously differentiable on the circle $\mathbb{T}$.
Using the fact that $\hat{f}(n)= \frac{\hat{f}'(n)}{in}$, Cauchy-Schartz and Parseval's identifty, I get $$\left|\sum \hat{f}(n)\right| < \frac{\pi}{\sqrt3} \sqrt{\frac{1}{2 \pi} \int_{\mathbb{T}}|f'(t)|^2\,\mathrm{d}t}$$
But this last integral is the norm of $f'$ in the $L_2$ space. How do we know $f'$ is in fact in that space? Is it because $f$ is continuously differentiable?