# Supnorms are equivalent with respect to two different bases

If $$F$$ is a finite dimensional vector spaces over $$\mathbb R$$, I have to show that suprimum norm with respect to two different bases are equivalent.

I know how to prove that in a finite dimentional vector space any two norms are equivalent. In that proof we use induction on dimention.

Edit: without using that (in finite dimentional spaces two norms are equivalent) How can one prove this.

I tried to do this but no result. Any help

• If you know that any two norms are equivalent then there is nothing left to prove. – Kavi Rama Murthy Dec 22 '19 at 12:31
• Yes but iwnt to prove this without using that . – thomson Dec 22 '19 at 12:34
• See this answer for proving that two norms on a finite-dimensional vector space are equivalent: math.stackexchange.com/a/599866/38584 – Math1000 Dec 22 '19 at 13:00

Let $$B = \{b_1, \ldots, b_n\}$$ and $$C = \{c_1, \ldots, c_n\}$$ be two bases for $$F$$ which define norms $$\left\|\sum_{i=1}^n \beta_ib_i\right\|_B := \max_{1\le i \le n}|\beta_i|, \qquad \left\|\sum_{i=1}^n \gamma_ic_i\right\|_C := \max_{1\le i \le n}|\gamma_i|$$ For $$x := \sum_{i=1}^n \beta_ib_i \in F$$ we have
$$\left\|\sum_{i=1}^n \beta_ib_i\right\|_C \le \sum_{i=1}^n|\beta_i|\|b_i\|_C \le \left(\sum_{i=1}^n\|b_i\|_C\right)\left(\max_{1\le i \le n} |\beta_i|\right) = \left(\sum_{i=1}^n\|b_i\|_C\right)\left\|\sum_{i=1}^n \beta_ib_i\right\|_B$$
or $$\|x\|_C \le \left(\sum_{i=1}^n\|b_i\|_C\right)\|x\|_B$$.
Analogously we get $$\|x\|_B \le \left(\sum_{i=1}^n\|c_i\|_B\right)\|x\|_C$$ so we conclude that the norms $$\|\cdot\|_B$$ and $$\|\cdot\|_C$$ are equivalent.