Riesz theorem

Riesz's theorem, in a mathematical context, relates to the behavior of linear functionals on certain function spaces. It states that every continuous linear functional on a space of integrable functions (specifically, \( L^p \) spaces for \(1 < p < \infty\)) can be represented as an integral against a fixed function. In simple terms, this means that any way of consistently assigning a number to each function—following linearity and continuity—can be thought of as measuring the function by integrating it with a specific fixed "weight" function. This provides a deep connection between abstract functionals and concrete integrals.