how to prove vector norm equivalence in finite dimensional space($\mathbb{R}^{n}$)? In most of the vector norm material, it was mentioned that the following inequalities can be proved, but no one provided the proof:
$$\lVert x\rVert_2\le\lVert x\rVert_1\le\sqrt{n}\lVert x\rVert_2;$$ 
$$\lVert x\rVert_\infty\le\lVert x\rVert_2\le\sqrt{n}\lVert x\rVert_\infty;$$  $$\lVert x\rVert_\infty\le\lVert x\rVert_1\le n\lVert x\rVert_\infty.$$
Is that very easy to prove this inequality?
Also wanted to know when the equality is attained?
 A: The most important norms are the three you mentioned. To prove the stated inequalities you have to find out how the unit sphere with respect to each of these norms looks like. All three spheres are symmetric with respect to reflections of the form $x_k\mapsto -x_k$, $1\leq k\leq n$. Therefore it is enough to look at the first "octant" where all $x_k$ are $\geq0$. In this "octant" we have
$$\|x\|_1=x_1+\ldots +x_n\>,\quad \|x\|_2=\sqrt{x_1^2+\ldots+ x_n^2}\>,\quad\|x\|_\infty=\max_k\ x_k\ .$$
It follows that the unit spheres belonging to these norms are an $n$-dimensional "octahedron", a euclidean $n$-ball of radius $1$, and an $n$-dimensional cube of side-length $2$. Given this interpretation I can leave it to you to prove the stated inequalities.
A: For the first, one direction is Holder's and the other is pretty simple (remember that $(x+y)^2=x^2+y^2+2xy$ and what this implies when $x,y\ge 0$).  The other two are pretty simple if you write out what the individual terms mean.  In fact, the third follows directly from the first two.
As for equality, Holder's inequality has specific conditions for equality, and the rest only require looking at how the inequalities were shown.
A: The first inequality is the tricky part.
 Considering the relation between Root-Mean Square-Arithmetic Mean, rms>= Am, you can figure it out. Other inequalities, you can find out if you write the definitions out.
The following link maybe helpful:
http://www.artofproblemsolving.com/Wiki/index.php/Root-Mean_Square-Arithmetic_Mean-Geometric_Mean-Harmonic_mean_Inequality
