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Volvieras

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consider a function f(x). the inner product <f|f> = int [ |f(x)|^2 dx] which is zero only* when f(x) = 0

only points to footnote 6, where Griffith points out: "what about a function that is zero everywhere except a few isolated points? the integral would still vanish, although the function does not. if this bothers you, you should have been a math major"

now i actually am a math major, and am interested in the reasoning behind this.

does it have to do with the fact that you would be integrating the value of some function over a tiny amount of dx, which would then be somehow equated to zero in the limit that dx -> 0?

if that is the case, how does this hold up with the fact that the integral of the kroneker delta function is one? i.e int[delta(x-a) * f(x)] = f(a)