First of all, my apology for not being able use Latex efficiently in this website. Please feel free to ask me for clarification if there is anything unclear in my question.
In a normed vector space on $\mathbb{R}$, $(X, \|\cdot\|)$, let $C(X)$ be the set of all continuous real-valued functions on $X$. It is not hard to check that all the norms which are equivalent to $\|\cdot\|$ will be continuous in $(X, \|\cdot\|)$. Let $\|\cdot\|_{op}$ denote the operator norm. Given a function $f$ in $C(X)$, $\|f\|_{op}$ = $\sup${|$f(x)$|:$\|x\|$ = 1}. Now my question is that: Inside the vector space $(C(X), \|\cdot\|_{op})$, does the set of all other norms continuous in $(X, \|\cdot\|)$ have any properties worth noticing?
I check a few books and can not find any information related to this. I wonder if the set of norms mentioned above, for instance, closed in $(C(X), \|\cdot\|_{op})$. When $X$ has finite dimension, $X$ has only one equivalent class of norms so that all of them will be continuous. When $X$ has infinite dimension, will each equivalent class of norms has the same properties as others do?
Any responses will be appreciated.
P.S.: A big thank for copper.hat's help :)