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 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() 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 = \int x(t)*y(t) dt,
where the discrete measure is of course used on S.
dave