Riesz representation theorem
The Riesz representation theorem reveals that, in a Hilbert space, there is no other way to measure a vector linearly and continuously except by taking its inner product with some fixed vector. Every continuous “scalar reading” of vectors is secretly “dot with y” for one and only one y. The dual space, which could a priori be a strange new object, turns out to be a faithful copy of the space itself.
Stated precisely: if H is a Hilbert space and f : H -> scalars is a bounded (continuous) linear functional, then there exists a unique y in H such that f(x) = <x, y> for all x in H, and moreover the functional's norm equals the representing vector's length, ||f|| = ||y||. The correspondence f <-> y is a bijection between the dual space H* and H; over the reals it is a linear isometric isomorphism, over the complex field it is conjugate-linear.
The proof is a clean application of the projection theorem. If f is identically zero take y = 0; otherwise the kernel of f is a closed subspace of codimension one, its orthogonal complement is one-dimensional, and scaling a unit vector from that complement produces exactly the representing y. This is why a Hilbert space is called self-dual, a property that fails for general Banach spaces.
On l^2 take the functional f(x) = sum over n of x_n / 2^n. It is bounded since |f(x)| <= ||x|| ||y|| with y = (1, 1/2, 1/4, ...), and Riesz says f(x) = <x, y> for exactly this y. Its norm is ||f|| = ||y|| = sqrt(sum 1/4^n) = sqrt(4/3) = 2/sqrt(3).
A concrete functional on l^2 and the unique vector representing it via the inner product.
This theorem underlies the bra-ket notation of quantum mechanics: every bra <psi| is the continuous functional “inner product with the ket |psi>.” It is also the foundation of weak formulations of differential equations, where solutions are characterized by how they pair against test functions.