Prove
$\displaystyle ||x||_{2}=(\sum^n_{i=1}|x_{i}|^2)^{1/2}$
Do i need to use triangle inequality or Cauchy Schwartz inequality ?
Where i need to start ?
I could be wrong, but that equation is usually taken as the definition of the $\displaystyle \ell_{2}$ norm, at least for the finite-dimensional case. In that case, you wouldn't need to prove it. Could you please provide a little more context?
I see what you mean now. You will need to prove the following:
1. $\displaystyle \|a x\|_{2}=|a|\cdot\|x\|_{2}$,
2. $\displaystyle \|x+y\|_{2}\le\|x\|_{2}+\|y\|_{2}$, and
3. $\displaystyle \|x\|_{2}=0\;\leftrightarrow x=0$,
for the norm definition you gave in the OP and for all vectors $\displaystyle x$ and $\displaystyle y$ and scalars $\displaystyle a$.
Does that get you started?
What? You have posted no work. How could we know if you're doing it right? THis problem isn't too bad if you just write down the actual formulas.
P.S. The upper bound in the norm should be infinity if we are, in fact, dealing with $\displaystyle \ell_2$ which is the set of all square summable sequences