If H is a real Hilbert space we denote by H∗ its dual – i.e., H∗ is the set of continuous linear forms on H. If h ∈ H then the mapping v → (h, v) (1.13) is an element of H∗. Indeed this is a linear form that is to say a linear mapping from H into R and the continuity follows from the Cauchy–Schwarz inequality |(h, v)| ≤ |h| |v|. (1.14) The Riesz representation theorem states that all the elements of H∗ are of the type (1.13) which means can be represented by a scalar product. This fact is easy to see on Rn. We will see that it extends to infinite-dimensional Hilbert spaces. First let us analyze the structure of the kernel of the elements of H∗.
(1) It is said that the mapping v→(h,v) is an element of H* because it is a linear form from H to ℝ, satisfying the definition of an element of H* as a space consisting of all continuous linear forms on H.
(2) If |h*(v)|=|(h,v)|≦|h| |v| holds, then it implies that h* is a continuous linear form because the Cauchy-Schwarz inequality ensures that the value of h*(v) is bounded by a multiple of the norm of h and v. Thus, h* is uniformly continuous on H, and therefore, a continuous linear form on H.