I am seeking for some intuition why norm (for any reasonable norm on functions) of a function is smaller if the function is smoother.
2026-03-26 19:27:07.1774553227
Norm of a function, Smoothness Penalization
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Norms may not necessarily be related smoothness in any way.
The uniform norm $\|f\|_u=\sup_{x \in [0,1]} |f(x)|$ on the space of continuous functions $C[0,1]$ is unrelated to smoothness. There are nowhere differential functions of arbitrarily small uniform norm.
But on the space of $L_2[0,1]$ absolutely continuous functions the norm $\|f\|=\int_0^1|f(t)| \ dt+\int_0^1 |f'(t)| \ dt$ does measure smoothness in some sense.