On Equivalent Norms in an Infinite Dimensional Vector Space How many non-equivalent norms can we define in an infinite dimensional vector space? Is there any explicit expression?
 A: There are plenty of non-equivalent norms. Let $X$ be an infinite-dimensional normed space with norm $\|\cdot\|_X$. 
Let $Y$ be another normed space with norm $\|\cdot\|_Y$. Let $T\in \mathcal L(X,Y)$ be compact and injective. Then 
$$
\|x\|_T:=\|Tx\|_Y
$$
is a norm on $X$. Moreover, $\|x\|_T \le \|T\|_{\mathcal L(X,Y)} \|x\|_X$. However, both norms cannot be equivalent: If there would be a constant $c>0$ such that
$$
\|x\|_X \le c\|x\|_T = c\|Tx\|_Y \quad\forall x\in X,
$$
this would imply that $T^{-1}$ is a continuous operator from $Im(T)$ to $X$, which is a contradiction (as compact operators cannot have continuous inverses).
A: There are exactly $2^{\dim X}$ inequivalent norms on an infinite dimensional vector space. You can find the proof here. 
A: When you are given just a vector space $X$, then you have as many non-equivalent norms on $X$ as many non-isomorphic normed spaces you can find with the same linear dimension. This is the standard `structure transport' argument. For instance, suppose that you are given a vector space of dimension continuum. Then for each infinite-dimensional Banach space $Y$ of linear dimensional dimension $\mathfrak{c}$ you can find a linear bijection $T_Y\colon X\to Y$ which gives you a norm $\|x\|:=\|T_Yx\|$. 
