Riesz representation theorem

A linear functional on V is a linear map phi : V -> F that eats a vector and returns a scalar — the rows of a matrix, dual vectors, the things in V*. The Riesz theorem makes a startling claim: on a finite-dimensional inner product space, EVERY such functional is secretly just take-the-inner-product-against-a-fixed-vector.

Precisely: for each linear functional phi there is a unique vector v in V with phi(u) = <u,v> for all u. The vector v is the functional in disguise. To build it, take an orthonormal basis e_1,...,e_n and set v = sum conj(phi(e_k)) e_k; one checks <u,v> = phi(u) directly.

This gives a canonical identification of V with its dual V*, with no choice of basis required — only the inner product. (The map phi -> v is conjugate-linear in the complex case, which is why people sometimes call it an antilinear isomorphism.) The inner product is exactly the structure that lets a space see its own dual.

Riesz is the engine behind the adjoint: to define T^*, you fix v, note that u -> <T u, v> is a functional, and Riesz hands you the unique vector T^* v representing it. In infinite dimensions the theorem still holds for Hilbert spaces with bounded functionals — the cornerstone of functional analysis.