$T(n) = 2 T \left ( \sqrt n \right ) + n$
Substitution for $n$:
Let $n = 2^r$. Thus, $r = \log_2 n$. Putting this value of $n$ we get
$T \left ( 2^r \right ) = 2 T \left ( 2^{r/2} \right ) + 2^r$
Substitution for $T(\cdot)$:
Let $S(r) = T \left ( 2^r \right )$.
Thus,
$S(r) = 2 S \left (\frac{r}{2} \right ) + 2^r$
Use Master Method for $S(r)$
$S(r) = \mathcal O(2^r)$
Substitute back to get solution for $T(\cdot)$
$\begin{align}
T \left ( 2^r \right ) &= S(r) = \mathcal O(2^r) \\
T(r) &= S(\log_2 r) = \mathcal O(2^{\log_2 r}) \\
T(r) &= \mathcal O(r)
\end{align}$