Is space $C[0,1]$ with norm $\parallel f \parallel=\max|f(x)|$ (space of continuous functions on $[0,1]$) isomorphic to space $C^1[0,1]$ with norm $\parallel f \parallel=\max|f(x)|+\max|f'(x)|$ (space of continuously differentiable functions on $[0,1]$) ? Under isomorphism I mean continuous linear bijective operator between these two spaces (and also the inverse is continuous). If yes, is there any explicit example of such isomorphism ? Thank you very much for Your answers.
[Math] isomorphism between $C[0,1]$ and $C^1[0,1]$
banach-spacesfunctional-analysis
Best Answer
As discussed in the comments, this boils down to showing that $C[0,1]\oplus\mathbb C\cong C[0,1]$. This works because $C[0,1]$ has a Schauder basis $g_n$; in fact, I want to work with specifically the Faber-Schauder basis. With respect to this basis, partial sums (up to $N=2^n$) of an expansion $f=\sum a_n g_n$ are piecewise linear interpolations of $f$ at the points $k2^{-n}$, and the coefficients $a_n$ are given by these values of $f$. In particular, $\|f\|=\sup |a_n|$.
It is now clear that $(f,a)\mapsto ag_0 + \sum a_n g_{n+1}$ has the desired properties.