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:
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.