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  2. Measurable function - Wikipedia

    en.wikipedia.org/wiki/Measurable_function

    A Lebesgue measurable function is a measurable function : (,) (,), where is the -algebra of Lebesgue measurable sets, and is the Borel algebra on the complex numbers. Lebesgue measurable functions are of interest in mathematical analysis because they can be integrated.

  3. Square-integrable function - Wikipedia

    en.wikipedia.org/wiki/Square-integrable_function

    In mathematics, a square-integrable function, also called a quadratically integrable function or function or square-summable function, [1] is a real- or complex-valued measurable function for which the integral of the square of the absolute value is finite.

  4. Lp space - Wikipedia

    en.wikipedia.org/wiki/Lp_space

    In mathematics, the L p spaces are function spaces defined using a natural generalization of the p-norm for finite-dimensional vector spaces.They are sometimes called Lebesgue spaces, named after Henri Lebesgue (Dunford & Schwartz 1958, III.3), although according to the Bourbaki group (Bourbaki 1987) they were first introduced by Frigyes Riesz ().

  5. Locally integrable function - Wikipedia

    en.wikipedia.org/wiki/Locally_integrable_function

    Measure and integration (as the English translation of the title reads) is a definitive monograph on integration and measure theory: the treatment of the limiting behavior of the integral of various kind of sequences of measure-related structures (measurable functions, measurable sets, measures and their combinations) is somewhat conclusive.

  6. Riesz–Fischer theorem - Wikipedia

    en.wikipedia.org/wiki/Riesz–Fischer_theorem

    The Riesz–Fischer theorem also applies in a more general setting. Let R be an inner product space consisting of functions (for example, measurable functions on the line, analytic functions in the unit disc; in old literature, sometimes called Euclidean Space), and let {} be an orthonormal system in R (e.g. Fourier basis, Hermite or Laguerre polynomials, etc. – see orthogonal polynomials ...

  7. L-infinity - Wikipedia

    en.wikipedia.org/wiki/L-infinity

    is a function space.Its elements are the essentially bounded measurable functions. [2]More precisely, is defined based on an underlying measure space, (,,). Start with the set of all measurable functions from to which are essentially bounded, that is, bounded except on a set of measure zero.

  8. Convergence in measure - Wikipedia

    en.wikipedia.org/wiki/Convergence_in_measure

    If X = [a,b] ⊆ R and μ is Lebesgue measure, there are sequences (g n) of step functions and (h n) of continuous functions converging globally in measure to f. If f and f n (n ∈ N) are in L p (μ) for some p > 0 and (f n) converges to f in the p-norm, then (f n) converges to f globally in measure. The converse is false.

  9. Hilbert space - Wikipedia

    en.wikipedia.org/wiki/Hilbert_space

    The problem is a differential equation of the form [()] + = for an unknown function y on an interval [a, b], satisfying general homogeneous Robin boundary conditions {() + ′ ′ = + ′ ′ =. The functions p, q, and w are given in advance, and the problem is to find the function y and constants λ for which the equation has a solution.