# Functional Analysis: Compactness of Ac:={f∈C1([0,1],ℝ):∫10|f(x)|2dx+∫10|f′(x)|2dx≤c}

#### SemperFi

##### New member
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!

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