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 :)

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    $\begingroup$ It is not true that all norms on $X$ are continuous. I'm also not sure what you mean by the "operator norm" on $C(X)$. $\endgroup$ – Eric Wofsey Sep 9 '18 at 3:44
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    $\begingroup$ When you write all the norms are continuous, you need to specify what you mean by continuous. Every norm is Lipschitz continuous with respect to the topology it induces, but not necessarily with respect to another norm. $\endgroup$ – copper.hat Sep 9 '18 at 4:50
  • $\begingroup$ Thanks for copper.hat's clarification. I agree that a norm might not be continuous in a topology induced by another norm. I am sure that inside (X, ||.||_a), any other norms equivalent to ||.||_a will be continuous. Let me edit the question now. $\endgroup$ – Sanae Kochiya Sep 9 '18 at 15:23
  • $\begingroup$ $\|f\|_{op}$ is not finite for all $f\in C(X)$. Do you mean to consider only the subspace of $C(X)$ of functions of finite norm? $\endgroup$ – Eric Wofsey Sep 9 '18 at 15:51
  • $\begingroup$ @EricWofsey In fact, I can not tell for sure whether $\|g\|_{op}$ is always finite given that g is a norm other than $\|\cdot\|$ and is equivalent to $\|\cdot\|$. We could focus only on all bounded continuous real-valued functions on X. Within one equivalent class of norms, I am not sure whether the supremum of all of their operator norm will be finite or not. If we focus on those norms with finite $\|\cdot\|_{op}$, their behaviors in the set of all bounded continuous real-valued functions might reflect some properties of the equivalent class each of them belong to. $\endgroup$ – Sanae Kochiya Sep 10 '18 at 21:44

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