What is proof of Basic Hilbert Transform?
Hilbert transform has the definition that its CTFT is:

$$
H(\Omega) = 
\begin{cases}
- \frac{\pi}{2}   &amp;,&amp; \text{ for } \Omega &gt; 0 \\
+ \frac{\pi}{2}   &amp;,&amp; \text{ for } \Omega &lt; 0 \\
\end{cases}
$$

from which you define the time-domian impulse response $h(t)$ associated Hilbert transformer from the inverse CTFT as:

$$ h(t) = \mathcal{IFT} \{ H(\Omega) \}  = \frac{1}{2\pi} \int_{-\infty}^{\infty} H(\Omega) e^{j \Omega t} d\Omega $$

You can evaluate the above integral or just use CTFT properties and pairs to conclude that:
$$ \boxed{ h(t) = \frac{1}{\pi t} } $$

Note that the associated impulse response $h(t)$ of the Hilbert transfomer is noncausal, two-sided and unstable and its Fourier transform can be expressed alternately as
$$ H(\Omega) = -j \text{sgn}(\Omega) $$
where $\text{sgn}(\cdot)$ is the sign function.

Hilbert transformer is most useful for defining and generating the analytic signal $x_+(t)$ associated with the signal $x(t)$, with has the property that 

$$
X_+(\Omega) = 
\begin{cases}
2 X(\Omega)  &amp;,&amp; \text{ for } \Omega &gt; 0 \\
0            &amp;,&amp; \text{ for } \Omega &lt; 0 \\
\end{cases}
$$