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What you write is blatently wrong (very unusual at least). Two functions are equal, if their graphs are equal. What happens here is that the vector space N of Null-functions (functions whose support is a measure-0-set) must be (is) factored out from the space of sq-integrable functions cL2 (curlyL).
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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
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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
592:"An equivalent definition is to say that the square of the function itself (rather than of its absolute value) is Lebesgue integrable. For this to be true, the integrals of the positive and negative portions of the real part must both be finite, as well as those for the imaginary part."
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Thus L2=cL2/N. The vector space of classes f+N, f sq. int. (f in cL2), then becomes a
Hilbert space. This is correctly described in the article about L2 on WP. What you describe as "new equality" is equality in the quotient space cL2/N.
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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
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value of a measerable function. Not using the positive and negative parts of Re(f) and Im(f). Thus the (mathematically correct, but too technical and trivial) section
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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.
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metrics under the revised equality. For more details see Naylor and Sell, Linear
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Two functions may attain different values on a set of measure zero and still be considered equal under any
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until a consensus is reached, and readers of this page are welcome to contribute to the discussion.
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integrable is defined via finite
Lebesque integral of the <|absolute|: -->
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https://en.wikipedia.org/Lebesgue_integration#Lebesgue_integrable
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which is much better written then this article here.
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these functions when you define L spaces, right? ---
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