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I hope this is not an obviously stupid question, I'm quite tired and hence extra slow today. I don't understand the following proof: enter image description here

Given the context I think $X = L^2 (\mathbb T)$, so $T_n : L^2 (\mathbb T) \to \mathbb R$. (The context is: Fourier series and uniform boundedness principle)

I don't see how we get the last line from the penultimate one. The operator norm is the $\sup$ over $f$ with norm equal to $1$. But the domain is $L^2$ so it comes with the $L^2$ norm. I'm aware that $\|f\|_1 \leq \|f\|_2$ and that I can apply Hölder to either get $\|T_n\| \leq \|f\|_2 \|D_n\|_2 = \|D_n\|_2$ or $\|T_n\| \leq \|f\|_\infty \|D_n\|_1$. But I want $\|T_n\| \leq \|D_n\|_1$.

Thanks for your help.

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    @t.b. Thank you very much.2012-08-11

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The domain of $T_n$ cannot be $L^2$ for the proof to work. There are $f\in L^2$ such that $\|f\|_{\infty}=\infty$ and then the proof is flawed.