Show that $L^2 ([ a, b])$ is the only inner product space among the spaces $L^P([a, b])$.

Asked Dec 17 '14 at 07:19

Active Dec 17 '14 at 07:27

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I know that $L^2 ([ a, b])$ is an inner product space with parallelogram law. But I cannot prove it for $p \neq 2$.

edited Dec 17 '14 at 07:27

MathMajor

asked Dec 17 '14 at 07:19

ali f

2

Did you try to see if the parallelogram law holds for $p\neq 2$? – 5xum Dec 17 '14 at 07:31
Excuse Me.I understand that the parallelogram law does not hold for p≠2. In general, how does this show? – ali f Dec 17 '14 at 12:17
Well, for a start, you can try to see why the parallelogram law does not hold, i.e. try to prove it and see where you get stuck. – 5xum Dec 17 '14 at 18:59
Your answer here: https://math.stackexchange.com/questions/1302099/lp-space-not-having-inner-product – fairytale Jul 29 '17 at 19:31

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