What are the maximal ideals of $C^1[0,1]$?
We know that the maximal ideals of $C[0,1]$ are of the form $\{f:f(x)=0\}$ and we use the compactness of $[0,1]$ to prove this,but how do we find maximal ideals of $C^1[0,1]$?
This question is asked in our course on Banach Algebras.
This is related to this question. Maximal ideals in the algebra of continuously differentiable functions on [0,1]
In fact, by using the same method, we can prove that the maximal ideal space of $C^k[0,1]$ is $[0,1]$. The key point here is polynomials are dense in $C^k[0,1]$ for every integer $k$.