Riesz representation theorem
Introduction
The Riesz representation theorem is a fundamental result in functional analysis, a branch of mathematics that studies vector spaces and operators acting upon them. Named after Hungarian mathematician Riesz Frigyes, the theorem provides a representation of linear functionals in terms of measures, facilitating the study of dual spaces and the development of measure theory.
Statement of the Theorem
The Riesz representation theorem can be stated in various forms depending on the context. Here, we present the theorem in the context of Hilbert spaces, a complete inner product space that is a key object of study in functional analysis.
Let H be a Hilbert space, and let L be a bounded linear functional on H. Then, there exists a unique y in H such that for all x in H,
L(x) = <x, y>
where <., .> denotes the inner product in H. The vector y is called the Riesz representative of L.
Proof of the Theorem
The proof of the Riesz representation theorem involves several steps, including the construction of the Riesz representative and the verification of its properties. The proof relies on key concepts from functional analysis, such as the Cauchy–Schwarz inequality and the Baire category theorem.
Applications
The Riesz representation theorem has numerous applications in mathematics and physics. In mathematics, it is used in the study of partial differential equations, Fourier analysis, and quantum mechanics. In physics, it plays a crucial role in the mathematical formulation of quantum mechanics, particularly in the study of quantum states and operators.