The Method of Orthonormal Series. Example

Abstract

Consider, as usual, a Hilbert space H and an operator A which is positive definite on a linear set D _A dense in H. Let H _A be the space constructed in Chap. 10 with the inner product (u, v) _A which is, as we know, an extension of the inner product (u, v) _A defined originally on D _A by the relation

$${\left( {u,v} \right)_A} = \left( {Au,v} \right),\,\,\,\,u \in {D_A},\,\,\,v \in {D_A},$$

(12.1)

to the entire space H _A . In the preceding chapter, it was shown that the functional

$$Fu = {\left( {u,u} \right)_A} - 2\left( {f,u} \right),\,\,\,\,\,u \in {H_A},$$

(12.2)

assumes its minimum in H _A for a certain element u₀ uniquely determined by the element f from the relation

$${\left( {{u_0},u} \right)_A} = \left( {f,u} \right)\,\,\,for\,every\,\,\,u \in {H_A}.$$

(12.3)