Dear Colleagues,

Could you please help me in solving the following problem:

Let $\displaystyle X$ be the space of all $\displaystyle x$ such that $\displaystyle x=(x_{1}, x_{2}, ..., x_{n})$, and $\displaystyle x_{1}, x_{2}, ..., x_{n}$ are numbers, and $\displaystyle ||.||$ be any norm on $\displaystyle X$, and define the norm $\displaystyle ||.||_{2}$ on $\displaystyle X$ to be $\displaystyle ||x||_{2}=(|x_{1}|+|x_{2}|+...+|x_{n}|)^{1/2}$. We want to show directly that there is $\displaystyle a>0$ such that $\displaystyle a||x||_{2}\leqslant ||x||$ for any $\displaystyle x\in X$, but without using the fact that all norms on a finite dimensional vector space are equivalent. Instead we want to use the fact that " A continuous mapping of a compact subset $\displaystyle M$ of a normed space $\displaystyle X$ into the real numbers $\displaystyle R$ assumes a maximum and a minimum at some points of $\displaystyle M$ ".

I think the continuous mapping in my problem will be the norm but which norms will be used and what is the compact set.

Best Regards.

Raed.