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.

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.

Related concepts

Inner product space

Inner product space generalizes Euclidean geometry

Spectral theorem

Spectral theorem applies to normal operators on Hilbert spaces

Normed vector space

A Banach space is a complete normed vector space

Cayley–Hamilton theorem

Cayley-Hamilton theorem states every matrix satisfies its own characteristic equation

Cholesky decomposition

Cholesky decomposition factors A = LL^T for symmetric positive definite matrices

the L1 norm is not differentiable at zero

The L1 norm is not differentiable at zero because the absolute value function has a kink at zero

One email a day: 5 concepts + the 5 stories that matter →

Swipe through 100 ML concepts daily

Open TickerNews