Riesz representation theorem

The theorem states that every continuous linear functional on a Hilbert space is uniquely expressible as inner product with a fixed vector. Its converse holds as well: each vector defines such a functional.

Hilbert space: A complete inner-product space, where every Cauchy sequence converges to an element of the space. Completeness and the inner product provide the setting in which the representation holds.
Lax–Milgram theorem: A theorem guaranteeing unique weak solutions to coercive variational problems on Hilbert spaces. Its proof uses Hilbert-space representation to turn bounded forms into operators.
Normed vector space: A vector space equipped with a norm that assigns lengths to vectors. Continuity of the functional is defined using the norm on its domain.
Hahn–Banach theorem: A theorem extending bounded linear functionals from subspaces to entire normed spaces without increasing their norm. It applies broadly to normed spaces, while Riesz representation requires Hilbert-space structure.
Continuous linear functional: A linear map from a normed vector space to its scalar field that is continuous in the norm topology. This is the object the theorem represents by inner product with a vector.
Weak formulation: A formulation of a differential equation expressed through integral identities against test functions. The theorem identifies bounded test-function functionals with elements of the underlying Hilbert space.
Completeness (metric space): A metric space is complete when every Cauchy sequence converges to a point in that space. Completeness distinguishes Hilbert spaces from inner-product spaces where the theorem may fail.
Riesz representation theorem for locally compact spaces: A theorem identifying positive linear functionals on continuous functions with regular measures. This distinct theorem shares the name but represents functionals by measures, not Hilbert-space vectors.
Inner product: A positive-definite sesquilinear or bilinear form that induces a norm on a vector space. The representing vector acts through this pairing to produce the functional's values.
Green's function: A kernel that represents solutions of a linear differential equation in response to point sources. In suitable function spaces, representing functionals helps derive integral solution formulas.

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