From: rusin@vesuvius.math.niu.edu (Dave Rusin)
Newsgroups: sci.math
Subject: Re: 2-Norm
Date: 4 Apr 1998 05:00:21 GMT

alpha wrote:
> Could someone please explain what 2-norm function, denoted by ||  ||, 
> signifies

Dom D'Alessandro wrote:
> The Euclidean (or L2 norm) is
>
> ||V|| = sqrt(V . V)
>
> where (V . V) = the Euclidean inner product.

Amik stcyr  <stcyr@dms.umontreal.ca> wrote:
>  L2 norm ?????
>
>x in L2(R)
>
>                /
>||x||^2  = |  |x(t)| dt
>               /
>
>Pretty different than the Euclidian norm....

I think you mean  ||x||^2 = \int (x(t))^2 dt.

Actually they're not all that different. As with the second poster's
notation, this is indeed  ||x|| = sqrt(<x,x>)  where  < -- , -- >  is
the inner product on  L^2(R). Moreover, what we usually call  R^n  is
actually  L^2(n), that is, you can think of an n-tuple of numbers as
being a function defined on the discrete set  S = {1, 2, ..., n}; then the
dot product we use on  R^n  is precisely  <x, y> = \int x(t)*y(t) dt,
where the discrete measure is of course used on  S.

dave