No: If what you want is the L-infinity
norm, then go for it. A standard place
for that is numerical approximations of
special functions -- want guarantees on
the worst case error. And there is some
math to help achieve that. It's
sometimes called Chebyshev approximation.
But, in practice, the usual situation,
e.g., signal processing, multi-variate
statistics, there's no good reason
not to use L2 and many biggie reasons
to use it. E.g., for a given box
of data, commonly the better tools in
L2 just let you do better.
Or
to the customer: "If you will go for
a good L2 approximation, then we
are in good shape. If you insist
on L1 or L-infinity, then we will
need a lot more data and still
won't do as well.".
Again, a biggie example is just
classic Fourier series. Sure,
if you are really concerned about
the Gibbs phenomenon, then maybe
work on that. Otherwise, L2 is the
place to be.
E.g., L1 and L-infinity can commonly
take you into linear programming.
Generally you will be much happier
with the tools available to you
in L2.
Again, just now I just don't
have time for a more full,
complete, and polished explanation.
A really good explanation would
require much of a good ugrad
and Master's in math, with
concentration on analysis and
a wide range of applications.
I've been there, done that but
just don't have time to
write out even a good summary of
all that material here.
