Are there properties of vector space equipped with two norms?

Question

I am interested in a vector space equipped with two norms$ \lvert \lvert \cdot \rvert \rvert$ and $ \lvert \lvert \cdot \rvert \rvert ^*$ satisfies that there is $M>0$ such that $ \lvert \lvert x \rvert \rvert^* \leq M\lvert \lvert x \rvert \rvert $ for all $x\in X$. It is clear that any open set in $(X,\lvert \lvert \cdot \rvert \rvert^*)$ is open in $(X,\lvert \lvert \cdot \rvert \rvert)$. What conditions make an open set in $(X , \lvert \lvert \cdot \rvert \rvert )$ to be open in $(X , \lvert \lvert \cdot \rvert \rvert^* )$ ? I want to know that there are other relations something like that between both norms such as closed sets or linear map, aren't there? Could you suggest me about books or others?

Answer 1

Two norms, $||.||$ and $||.||^*$ on a vector space are said to be equivalent norms if there exists a constant $C > 0$ such that $\frac{1}{C}||x||\leq ||x||^* \leq C||x||$. It is a nice exercise to show that all norms on $\mathbb{R}^N$ are equivalent.