Riesz representation theorem

Riesz representation theorem connects Hilbert spaces with continuous dual spaces

The Riesz representation theorem establishes a fundamental connection between Hilbert spaces and their continuous dual spaces. This theorem demonstrates that every bounded linear functional on a Hilbert space can be represented as an inner product with a fixed vector in that space. This connection is crucial for understanding the structure and properties of Hilbert spaces and their duals.

Example

Consider the Hilbert space ℓ², consisting of all square-summable sequences. A bounded linear functional f on ℓ² can be represented as f(x) = ⟨x, y⟩, where y is a fixed sequence in ℓ². This inner product representation allows us to understand f in terms of the elements of ℓ².

Remember this

Understanding this theorem is essential for working with Hilbert spaces and their duals, as it provides a concrete way to represent linear functionals as inner products, simplifying many theoretical and practical applications.

Text adapted from Wikipedia, licensed under CC BY-SA 4.0.

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