Sign up ×  Given a measure space $X$ with its measure $\mu$, it
can be shown (I'll provide a proof if asked for) that
$\displaystyle \forall f \in L^\infty(X,\mu),~\textrm{if }
\exists p_0:\forall q \geq p_0, f\in L^q\cap L^\infty, \textrm{
then } \lim_{p \to \infty}\|f\|_p = \|f\|_\infty$ (whih by the
way justify this notation)  This convergence implies the
following: $\forall f, \forall \epsilon > 0, \exists q:=
q(f,\epsilon),\textrm{ such that } \forall p\geq q,
|\|f\|_p-\|f\|_\infty| < \epsilon$  This means that given an
approximation error of the infinity norm bounded by $\epsilon$,
I should be able to compute an (let's call it) index so that, I
don't need to go any further, but I have a priori knowledge of
the potential error.  The idea is, I am working on some pattern
recognition problems and I am using the infinity norm somewhere
there. However, as it is quite unreliable against outliers,
using a p-norm approximation allows to "average out" the local
outliers and get a more robust result. The higher t