If we have a continuous variable $x$ in $\mathbb R^n$. How do we compute norm of $x$ i.e $\|x\|$?
Say let $x = [\sin^2 x; \cos x]$ on some interval say $[0,t_f]$, so should we take $L_2$ norm of $\sin^2x$ and $L_2$ norm of $\cos x$ and then perform $\ell_2$ norm or is there another way to compute it?
It depends on which norm you want to compute. There are many possible ways: \begin{align*} &\int_0^{t_f} |\sin^2(t)| + |\cos(t)| \, \mathrm{d}t \\ &\int_0^{t_f} \sqrt{|\sin^2(t)|^2 + |\cos(t)|^2} \, \mathrm{d}t \\ &\int_0^{t_f} (|\sin^2(t)|^p + |\cos(t)|^p)^{1/p} \, \mathrm{d}t \\ &\Big(\big(\int_0^{t_f} |\sin^2(t)|\,\mathrm{d}t\big)^p + \big(\int_0^{t_f} |\cos(t)|\,\mathrm{d}t\big)^p\Big) ^{1/p} \end{align*} for any $p \in [1,\infty)$ and many more.