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Square-integrable function, the Glossary

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.^[1]

18 relations: Absolute value, Almost everywhere, Banach space, Cauchy sequence, Cauchy space, Complete metric space, Complex conjugate, Complex number, Equivalence class, Giovanni Sansone, Hilbert space, Inner product space, Integral, Lebesgue integral, Lp space, Mathematics, Measurable function, Real number.
Lp spaces

Absolute value

In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign.

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Almost everywhere

In measure theory (a branch of mathematical analysis), a property holds almost everywhere if, in a technical sense, the set for which the property holds takes up nearly all possibilities.

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Banach space

In mathematics, more specifically in functional analysis, a Banach space (pronounced) is a complete normed vector space. Square-integrable function and Banach space are functional analysis.

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Cauchy sequence

In mathematics, a Cauchy sequence is a sequence whose elements become arbitrarily close to each other as the sequence progresses.

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Cauchy space

In general topology and analysis, a Cauchy space is a generalization of metric spaces and uniform spaces for which the notion of Cauchy convergence still makes sense.

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Complete metric space

In mathematical analysis, a metric space is called complete (or a Cauchy space) if every Cauchy sequence of points in has a limit that is also in.

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Complex conjugate

In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign.

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Complex number

In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted, called the imaginary unit and satisfying the equation i^.

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Equivalence class

In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes.

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Giovanni Sansone

Giovanni Sansone (24 May 1888 – 13 October 1979) was an Italian mathematician, known for his works on mathematical analysis, on the theory of orthogonal functions and on the theory of ordinary differential equations.

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Hilbert space

In mathematics, Hilbert spaces (named after David Hilbert) allow the methods of linear algebra and calculus to be generalized from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Square-integrable function and Hilbert space are functional analysis.

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Inner product space

In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product.

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Integral

In mathematics, an integral is the continuous analog of a sum, which is used to calculate areas, volumes, and their generalizations.

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Lebesgue integral

In mathematics, the integral of a non-negative function of a single variable can be regarded, in the simplest case, as the area between the graph of that function and the axis. Square-integrable function and Lebesgue integral are lp spaces.

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Lp space

In mathematics, the spaces are function spaces defined using a natural generalization of the ''p''-norm for finite-dimensional vector spaces. Square-integrable function and Lp space are lp spaces.

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Mathematics

Mathematics is a field of study that discovers and organizes abstract objects, methods, theories and theorems that are developed and proved for the needs of empirical sciences and mathematics itself.

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Measurable function

In mathematics, and in particular measure theory, a measurable function is a function between the underlying sets of two measurable spaces that preserves the structure of the spaces: the preimage of any measurable set is measurable.

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Real number

In mathematics, a real number is a number that can be used to measure a continuous one-dimensional quantity such as a distance, duration or temperature.

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References

[1] https://en.wikipedia.org/wiki/Square-integrable_function

Also known as L2 Space, L2-function, L2-inner product, L2-space, L^2, L² space, Quadratic integrability, Quadratically integrable, Quadratically integrable function, Square integrability, Square integrable, Square integrable function, Square-integrability, Square-integrable, Square-integrable functions, Square-summable function.