How many non-equivalent norms can we define in an infinite dimensional vector space? Is there any explicit expression?
[Math] On Equivalent Norms in an Infinite Dimensional Vector Space
functional-analysisgeneral-topologymetric-spacesnormed-spacesvector-spaces
Best Answer
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).