Not all norms are equivalent in an infinite-dimensional space How to prove that not all norms are equivalent in an infinite-dimensional vector space?
In particular, I would like to prove that for a space $X$ of continuous real-valued functions defined on interval $[0,1]$, every two norms $\|\ .\|_p$ ($p \in [1, \infty]$) are not equivalent.
 A: Consider the two spaces $L^{p_1}(-1, 1), L^{p_2}(-1, 1)$ with $1\le p_1<p_2 < \infty$. Let $f \in L^{p_1}\cap L^{p_2}(-1, 1)$. For all $\lambda\ge 1$ define 
$$
f_\lambda(x)=\lambda^\frac1{p_1}f(\lambda x).$$
Then you have 
$$\tag{1}
\|f_\lambda\|_{p_1}=\|f\|_{p_1}
$$
and
$$\tag{2}
\|f_\lambda\|_{p_2} = \lambda^{\frac{p_2-p_1}{p_1p_2}}\|f\|_{p_2}$$ 
 If the two norms were equivalent on $L^{p_1}\cap L^{p_2}$ you would have, by definition, 
$$
c\|f_\lambda\|_{p_2} \le \|f_\lambda\|_{p_1}\le C\|f_\lambda\|_{p_2}$$
but (1) and (2) show that this is not possible (to see why, let $\lambda \to \infty$).
A: Simple example of two non-equivalent norms on infinitely-dimensional space:
Consider space of all contnuously differentiable functions $X = C^1 [0,1]$. Then equipping it with the norm:
$$ \|f \|_{C^1} = \sup \limits_{x \in [0,1]} |f| + \sup  _{x \in [0,1]} |f'| $$
gives us a complete space (Banach space), but if we consider norm:
$$ \|f \|_\infty = \sup \limits_{x \in [0,1]} |f| $$
Then the normed space is not complete, hence the norms are not equivalent.
Another way to see it is to consider sequence $f_n(x) = \frac 1n \sin (2 \pi  n x)$, which is convergent to $0$ function in $\| \cdot \|_\infty$:
$$ \| f_n \|_\infty = \frac 1n \rightarrow 0 $$
norm, but not convergent to $0$ in $ \| \cdot \|_{C^1}$:
$$ \| f_n \|_{C^1} = \frac 1n + 1$$
