To prove that, I considered the identity mapping $J$ and since unit ball in $C^1$ is compact in space $C$,, then it has finite dimension. But I have no idea how to find a finite basis of this space.
2026-05-05 23:22:49.1778023369
Space $C^1$ is finite dimensional in $C[a,b]$. What is its basis?
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$C^1$ is the space of continuously differentiable functions? It definitely is not finite dimensional. (For instance, the functions $1, x, x^2, x^3, \dots$ are all in $C^1$ and are linearly independent.)
The issue is that you're asking about compactness in a different topology. The theorem is that if $X$ is a vector space with a norm, such that the unit ball of that norm (or any set containing the ball) is compact in the topology induced by that same norm, then the space is finite dimensional. Here you are asking about compactness with respect to another norm. The $C$-closure of the $C^1$-ball is not compact in the $C^1$ norm, and it doesn't contain a ball of the $C$-norm. So there's no contradiction.