Functional Analysis: Compactness of Ac:={f∈C1([0,1],ℝ):∫10|f(x)|2dx+∫10|f′(x)|2dx≤c}
Hello everybody,
I would like to show that the closure of the set \(\displaystyle A_c:=\{f \in C^1([0,1], \mathbb{R}):\int_{0}^{1}|f(x)|^2dx+\int_{0}^{1}|f'(x)|^2dx \leq c\}\) for fixed \(\displaystyle c>0\) is compact in the space \(\displaystyle C([0,1], \mathbb{R})\). I would attack this problem using Arzela-Ascoli's theorem. For that, I need to show (totally) boundedness and equicontinuity. Obviously, if \(\displaystyle f \in A_c\), then both \(\displaystyle f\) and \(\displaystyle f'\) are bounded. However, is there a uniform bound for all \(\displaystyle f \in A_c\)? I'm stuck here. What other properties do functions in \(\displaystyle A_c\) have?
I'm grateful for any help!
Hello everybody,
I would like to show that the closure of the set \(\displaystyle A_c:=\{f \in C^1([0,1], \mathbb{R}):\int_{0}^{1}|f(x)|^2dx+\int_{0}^{1}|f'(x)|^2dx \leq c\}\) for fixed \(\displaystyle c>0\) is compact in the space \(\displaystyle C([0,1], \mathbb{R})\). I would attack this problem using Arzela-Ascoli's theorem. For that, I need to show (totally) boundedness and equicontinuity. Obviously, if \(\displaystyle f \in A_c\), then both \(\displaystyle f\) and \(\displaystyle f'\) are bounded. However, is there a uniform bound for all \(\displaystyle f \in A_c\)? I'm stuck here. What other properties do functions in \(\displaystyle A_c\) have?
I'm grateful for any help!
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