# Talk:Square-integrable function

Two functions may attain different values on a set of measure zero and still be considered equal under any ${\displaystyle L_{p}}$ metric. Only the ${\displaystyle L_{2}}$ metric is induced by an inner product, because the polarization identity does not hold for other values of ${\displaystyle p}$. Therefore the other ${\displaystyle L_{p}}$ 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 ${\displaystyle L_{p}}$ space can be a metric space and how an ${\displaystyle L_{2}}$ 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 ${\displaystyle L_{p}}$ 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)