Lesson plan
Square-integrable function
In mathematics, a square-integrable function, also called a quadratically integrable function or L^2 function or square-summable function, is a real- or complex-valued measurable function for which the integral of the square of the absolute value is finite. Thus, square-integrability on the real line (-\infty, +\infty) is defined as follows. One may also speak of quadratic integrability over bounded intervals such as [a,b] for a \leq b . 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. The vector space of (equivalence classes of) square integrable functions (with respect to Lebesgue measure) forms the L^p space with p = 2. Among the L^p spaces, the class of square integrable functions is unique in being compatible with an inner product, which allows notions like angle and orthogonality to be defined. Along with this inner product, the square integrable functions form a Hilbert space, since all of the L^p spaces are complete under their respective p -norms. Often the term is used not to refer to a specific function, but to equivalence classes of functions that are equal almost everywhere.
Article summary
In mathematics, a square-integrable function, also called a quadratically integrable function or L^2 function or square-summable function, is a real- or complex-valued measurable function for which the integral of the square of the absolute value is finite. Thus, square-integrability on the real line (-\infty, +\infty) is defined as follows. One may also speak of quadratic integrability over bounded intervals such as [a,b] for a \leq b . 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. The vector space of (equivalence classes of) square integrable functions (with respect to Lebesgue measure) forms the L^p space with p = 2. Among the L^p spaces, the class of square integrable functions is unique in being compatible with an inner product, which allows notions like angle and orthogonality to be defined. Along with this inner product, the square integrable functions form a Hilbert space, since all of the L^p spaces are complete under their respective p -norms. Often the term is used not to refer to a specific function, but to equivalence classes of functions that are equal almost everywhere.
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Discussion questions
- Summarise the article in your own words. Which single claim would you verify first, and why?
- Which claim in this article is most likely to be revised as the sources update? How would you check its current status?
- If two sources disagreed about one of these facts, what would you need in order to trust one over the other?
Bibliography
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Key claims and bibliography are generated from the same tracked claims and sources as the article page.