Talk:Square-integrable function

From Wikipedia, the free encyclopedia
Jump to: navigation, search

The article says, "The quadratically integrable functions form an inner product space." I'm pretty sure that isn't true! A function that differs from the zero function on a non-empty set of measure zero will have inner product zero, violating positive-definiteness. That's why you have to mod out by these functions when you define Lp spaces, right? ---Vectornaut (talk) 01:44, 29 May 2011 (UTC)

Two functions may attain different values on a set of measure zero and still be considered equal under any L_p metric. Only the L_2 metric is induced by an inner product, because the polarization identity does not hold for other values of p. Therefore the other L_p spaces are not inner product spaces. The equality of functions in the context of Lebesgue integration is redefined in terms of the Lebesgue integral of their difference. Therefore, even if they do differ on a set of Lebesgue measure zero, they are equal under the revised equality. This is how an L_p space can be a metric space and how an L_2 space is an inner product space. So the condition of sign definiteness (or positive definiteness, although that is redundant and can be derived by sign definiteness and the triangle inequality) holds for the L_p metrics under the revised equality. For more details see Naylor and Sell, Linear Operator Theory in Engineering and Science, Springer. — Preceding unsigned comment added by Helptry (talkcontribs) 03:21, 29 November 2012 (UTC)

Why does L2 norm redirect here!?[edit]

I'm looking for an explanation of L2 norm. I get here instead and I see L2 and I see norm, but if you are going to redirect me here then I would like to see L2 norm. — Preceding unsigned comment added by 108.80.114.110 (talk) 11:42, 10 July 2014 (UTC)

For what it's worth, L2 norm redirects to Norm (mathematics)#Euclidean norm, and L2-norm redirects to Square-integrable function. I think L2-norm should be fixed to redirect to Norm (mathematics)#Euclidean norm, too. – Tobias Bergemann (talk) 11:57, 10 July 2014 (UTC)