Any example of equivalent norms in infinite-dimensional vector space?

Question

I have read in this paper that the number of inequivalent norms in infinit-dimensional space is exactly $2^{dim X}$, In my guess if am true this mean that there are Equivalent norms in infinit dimensional space which I want to know one example of it ? and if there is no Equivalents norms in infinit dimensional vector space just any proof for that ?

Answer 1

If $\lVert\cdot\rVert$ is any norm on any vector space $X$, then $\lVert\cdot\rVert$ and $2\lVert\cdot\rVert$ are equivalent norms.

Answer 2

Here's a silly example: define on $X=\mathcal{C}^0([a,b],\mathbb{R})$ the norm $\lVert\cdot\rVert$ given by $\sup_{x\in [a,b]} f(x)=\lVert f\rVert.$ Define a second norm $\lVert\cdot\rVert'$ by $2\sup_{x\in [a,b]}f(x)=\lVert f\rVert'$. Then indeed we have that there exist constants $c$ and $C$ such that $$ c\lVert f\rVert'\le \lVert f\rVert\le C\lVert f\rVert'.$$ For instance, take $c=\frac{1}{2}$ and $C=1$. More generally, if we have a normed space over $\mathbb{R}$, a norm $\lVert\cdot\rVert$ determines a family of equivalent norms $\lambda \lVert\cdot\rVert$ for $\lambda\in \mathbb{R}$. This fact is independent of the dimensionality of the space.