Is there an explicit example of a complete norm on $C[0,1]$ that is not equivalent to $\|\cdot\|_\infty$?

Question

Does anyone have an constructive, explicit example for a norm $||.||$ on $C[0,1]$, such that $(C[0,1], ||\cdot||)$ is a Banach space, but such that $||\cdot||$ is not equivalent to $||\cdot||_{\infty}$? I know that if convergence in $||\cdot||$ implies pointwise convergence then it is equivalent to $||\cdot||_{\infty}$ (see https://math.stackexchange.com/q/4471871). There is also this result Finding norm on $C[0,1]$ , which is not equivalent to the supremum norm, but which still makes $C[0,1]$ into a separable Banach space, but it uses some isomorphism which is based on the existence of a Hamel basis, which one cannot explicit construct.

Answer 1

It is consistent, without the axiom of choice, that any linear operator between Banach spaces is continuous.

In that scenario, take the identity function between your two norms. It is linear, so it is continuous, but that means that the two norms are equivalent.

Answer 2

No, I believe such a thing cannot be "constructive, explicit".

What I know all about is the case where $(C[0,1], ||\cdot||)$ is a separable Banach space. Let's say "constructive, explicit" means its existence and its properties can be proved in ZF set theory. I.e.: not using the Axiom of Choice, Hahn-Banach theorem, or other things beyond ZF.

The following "automatic continuity" theorem is due to J.P.R. Christensen, early 1970's: If $A,B$ are complete separable metric groups, and the homomorphism $f : A \to B$ has the property of Baire, then $f$ is continuous.

! A set $E$ has the property of Baire iff $E = M \triangle U$ for some meager set $M$ and some open set $U$. A function $f : A \to B$ has the property of Baire means: for every open set $G$ in $B$, the inverse image $f^{-1}(G)$ has the property of Baire in $A$.

Shelah (1984) showed that it is consistent with ZF that every subset of a complete separable metric space has the property of Baire. In particular, every function between complete separable metric spaces has the property of Baire.

Suppose $\|\cdot\|$ is such a norm. Consider the identity map $i : (C,\|\cdot\|) \to (C,\|\cdot\|_\infty)$. By Shelah, it is consisten with ZF that $i$ has the property of Baire. Then, by Christensen, $i$ is continuous. Repeat with the two spaces interchanged. So $i$ is a homeomorphism.

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A similar question, with the same answer: is there a constructive, explicit isomorphism between the two groups $(\mathbb R,+)$ and $(\mathbb R^2,+)$.