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Does the norm $$\|f\|=\sup\limits_{t\in[0,T]}\int\limits^t_0|f(\tau)|\ d\tau$$ have a specific name?

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Yes. As $t \mapsto \int_0^t|f(\tau)|\, d\tau$ is monotone, the sup equals $\|f\|_1 = \int_0^T |f(\tau)|\,d \tau$, the $L^1([0,T])$-norm.

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Thank you very much for the answer! – Karla Sep 19 '12 at 12:31
So, more interesting would be $$\|f\|=\sup\limits_{t\in[0,T]}\left|\ \int\limits^t_0 f(\tau) \ d\tau \right|$$ – GEdgar Sep 19 '12 at 13:09

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