A theorem that characterizes continuous linear functionals on certain topological vector spaces by concrete objects: on a Hilbert space H every bounded linear functional is given by the inner product with a unique vector in H, and on C0(X) (continuous functions vanishing at infinity on a locally compact Hausdorff space X) every continuous linear functional is represented as integration against a unique regular signed (or complex) Borel measure.

Principle

Identify abstract dual elements with explicit geometric or measure-theoretic representatives so that duality becomes evaluation against a canonical object (a vector or a measure).

Demonstration

In L2([0,1]) each bounded linear functional L has a unique g in L2 with L(f)=∫_0^1 f(x) g(x) dx, equivalently L(f)=⟨f,g⟩. For C([0,1]) the dual is the space of finite signed regular Borel measures μ and L(f)=∫ f dμ.

Misapplication

Assuming the same inner-product representation holds in arbitrary Banach spaces such as Lp for p≠2, or assuming measure representation for all function spaces without local compactness or the C0 hypothesis.

Consequence

Transforms abstract dual-space questions into concrete analytic objects, enabling explicit constructions, norm calculations and spectral descriptions; it simplifies identification of adjoints and kernels.

Reversal

Elements of the algebraic dual that are not continuous, or continuous functionals on spaces lacking the required structure, cannot be written as inner products or integrals; the dual is then strictly larger or different.

Boundary

Applies as stated for Hilbert spaces and for C0(X) with X locally compact Hausdorff (or compact for C(X)); it does not extend verbatim to arbitrary Banach spaces, non-locally-compact X, or to discontinuous linear functionals.

Semantic Tension

The phrase 'Riesz representation' also labels related but distinct results (Riesz–Markov, Riesz lemma, Riesz representation in Hilbert spaces), so one must distinguish which representation (inner-product vs measure) is intended.

Synthesis

The Riesz representation theorem equates continuous linear functionals on key analytic spaces with concrete evaluators—vectors in Hilbert spaces or regular Borel measures on suitable topological spaces—thereby converting abstract duality into explicit integration or inner product evaluation.