The Galerkin Method

Abstract

Consider a separable Hilbert space H and a set M of its elements which is dense in H. According to Theorem 6.18, p. 79, if for some element u ∈ H

$$\left( {u,v} \right) = 0\,\,\,holds\,for\,every\,\,\,v \in M,$$

(14.1)

then it follows that u = 0 in H. Let now

$${\varphi _1},{\varphi _2},\,...$$

(14.2)

be a base in H. The assertion is that if (u, ϕ_k) = 0 holds for all k = 1,2,… then again u = 0 in H. Briefly written,

$$\left( {u,{\varphi _k}} \right) = 0\,\,\,for\,\,\,k = 1,2,\,... \Rightarrow u = 0\,\,\,in\,\,\,H.$$

(14.3)