Riesz–Markov–Kakutani representation theorem

The theorem identifies continuous linear functionals on spaces of continuous functions with integration against unique regular Borel measures, under specified conditions on the space and function space.

Riesz representation theorem for C₀(X): A representation theorem identifying bounded linear functionals on C₀(X) with finite regular Borel measures on a locally compact Hausdorff space X. This is the locally compact Hausdorff version, where functions vanish at infinity.
Locally compact Hausdorff space: A Hausdorff topological space in which every point has a compact neighborhood. This is the standard setting for the C₀(X) version.
Riesz representation theorem: A family of theorems representing linear functionals or vectors through measures or inner products, depending on the setting. The theorem is a central measure-theoretic member of this broader family.
Riesz representation theorem for Hilbert spaces: The theorem identifying each continuous linear functional on a Hilbert space with inner product against a unique vector. It represents functionals by vectors, not by measures on a topological space.
Riesz representation theorem for C(X): A representation theorem identifying bounded linear functionals on C(X) for compact Hausdorff X with finite regular Borel measures. Compactness lets the theorem apply to all continuous functions, including constants.
C₀(X): The space of continuous functions on a locally compact space that vanish at infinity. Its supremum norm makes bounded linear functionals the objects represented by measures.
Radon measure: A Borel measure on a suitable topological space that is locally finite and satisfies regularity properties. In common locally compact settings, the representing regular Borel measures are Radon measures.
Stone–Čech compactification: A compact Hausdorff space containing a completely regular space densely, with a universal extension property for bounded continuous functions. It connects bounded continuous functions on a noncompact space to continuous functions on a compact space.
Total variation measure: A positive measure that quantifies the full variation of a signed or complex measure. The norm of a represented functional equals the total mass of its representing measure's total variation.
Continuous linear functional: A linear map between topological vector spaces that is continuous with respect to their topologies. The theorem starts with such a functional on a space of continuous functions.

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