M. Riesz's theorem

Step 1 ($L^2$, directly). By Plancherel and the multiplier formula, $\|Hf\|_{L^2}^2 = \int |\widehat{Hf}(\xi)|^2\,d\xi = \int |\mathrm{sgn}(\xi)|^2|\hat f(\xi)|^2\,d\xi = \|f\|_{L^2}^2$, using $|\mathrm{sgn}(\xi)|=1$ for $\xi\ne0$ (a single point does not affect the integral). So $H$ is an isometry on $L^2(\mathbb{R})$, in particular bounded with $C_2=1$.

Step 2 (general $1<p<\infty$, forward reference). The full range of exponents does not follow from Step 1 by any soft argument. It is a genuine theorem of real-variable Calderón–Zygmund theory: the Hilbert transform's kernel $1/(\pi x)$ satisfies exactly the smoothness and cancellation conditions of a Calderón–Zygmund kernel, so the general singular-integral machinery developed in the next section applies to it. That machinery produces a weak-type $(1,1)$ bound from the $L^2$ bound proved in Step 1, and Marcinkiewicz interpolation between the weak-$(1,1)$ bound and the $L^2$ bound (then a duality argument for $p>2$) completes the proof for every $1<p<\infty$ — this half of the argument is carried out in full in the Calderón–Zygmund section immediately below, rather than repeated here.