Let g1; g2; . . . ; gm be linear functional on a linear space X, and let S ={x X : gj(x) = 0,

Let g1; g2; . . . ; gm be linear functional on a linear space X, and let
S ={x ˆˆ X : gj(x) = 0, j = 1,2,..., m}=

Suppose that f ^ 0 is another linear functional such that such that f (x)= 0 for every x e S. Show that

Figure 3.20
The Fredholm alternative via separation
1. The set Z . {f} x, - g1(x), g2.x. . . . -gm.x ˆˆ X} is a subspace of Y . Rm+1.
2. e0 = (1,0,0, €¢€¢€¢, 0) ˆˆ „œm+1 does not belong to Z (figure 3.20).
3. There exists a linear functional 0 and Ï•(z) = 0 for every z ˆˆ Z,
4. Let Ï• (y) = l λTy where λ = λ0 , λ1,......λm) ˆˆ Y = „œ(m+1). For every z ˆˆ Z.
5. λ0 > 0.
6. f (x). . ˆ‘mi=1 λigi (x); that is f is linearly dependent on g1g2 .......gm.

n kernel g/ f(x) eo 9(x)