all norms are not equivalent
Let $V$ be the vector space of continuous functions^{} $[1,1]\to \mathbb{R}$ that are differentiable^{} at $0$. Then we can define norms
$$\parallel f\parallel =\underset{x\in [1,1]}{\mathrm{max}}f,$$ 
and
$${\parallel f\parallel}^{\prime}=\parallel f\parallel +{f}^{\prime}(0).$$ 
It is not difficult to find a sequence of functions ${f}_{1},{f}_{2},\mathrm{\dots}$ in $V$ such that

1.
${f}_{k}^{\prime}(0)=k$ for $k=1,2,\mathrm{\dots}$,

2.
$\parallel {f}_{k}\parallel =1$.
Then $\parallel {f}_{k}\parallel =1$, and ${\parallel {f}_{k}\parallel}^{\prime}=1+k$, so there is no $C>1$ such that
$${\parallel f\parallel}^{\prime}\le C\parallel f\parallel \mathit{\hspace{1em}}f\in V,$$ 
and $\parallel \cdot \parallel $ and $\parallel \cdot {\parallel}^{\prime}$ cannot be .
Title  all norms are not equivalent^{} 

Canonical name  AllNormsAreNotEquivalent 
Date of creation  20130322 15:36:11 
Last modified on  20130322 15:36:11 
Owner  matte (1858) 
Last modified by  matte (1858) 
Numerical id  6 
Author  matte (1858) 
Entry type  Example 
Classification  msc 46B99 