Let $C^{1}[0,1]$ be the space of all real valued functions on $[0,1]$ which have continuous derivatives and define the norm by;
$\|f\|:=|f(0)|+\sup\limits_{0\leq t \leq 1}|f'(t)|$
Is it a Banach space?
Is this a Banach Space
functional-analysis
functional-analysis
Let $C^{1}[0,1]$ be the space of all real valued functions on $[0,1]$ which have continuous derivatives and define the norm by;
$\|f\|:=|f(0)|+\sup\limits_{0\leq t \leq 1}|f'(t)|$
Is it a Banach space?
Best Answer
To prove that your space is complete we will use the fact that $C[0,1]$ is complete. Consider a Cauchy sequence $(f_n)\subset C^1[0,1]$. By definition of $C^1$ this implies that $(f_n')$ is a Cauchy sequence in $C[0,1]$. As $C[0,1]$ is Banach there exists some $g\in C[0,1]$ such that $f_n'\to g$ in $C[0,1]$. Furthermore $(f_n(0))$ is also Cauchy, and must converge to some $f_0\in \mathbb C$, by the completeness of the field. Let us now define $f:[0,1]\to\mathbb C$ by $$f(t)=f_0+\int_0^tg(x)dx.$$ We claim that $f_n\to f$ in $C^1[0,1]$. This is simply because the fundamental theorem of calculus gives us $$\|f-f_n\|=|f_n(0)-f_0|+\sup_t|f'_n(t)-g(t)|.$$ The result then follows by the nature of $g$ and $f_0$.