Let $C^1[0,1]$ be space of all real valued continuous function which are continuously differentiable on $(0,1)$ and whose derivative can be continuously extended to $[0,1]$. For $f$ in that set define the norm of $f = \max{\{\|f(t)\|, \|f'(t)\|\}}$ where $t$ is in $[0,1]$. can you show this space is complete under this norm? and let a linear operator $T:C^1[0,1] \longrightarrow C[0,1]$ defined by $T(f)=f'$, show that $T$ is continuous and norm of $T=1$.
Space of all continuously differentiable functions
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functional-analysis
banach-spaces