Let g1; g2; . . . ; gm be linear functional on a linear space X, and
Question:
S ={x 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 X} is a subspace of Y . Rm+1.
2. e0 = (1,0,0, ¢¢¢, 0) m+1 does not belong to Z (figure 3.20).
3. There exists a linear functional 0 and Ï(z) = 0 for every z Z,
4. Let Ï (y) = l λTy where λ = λ0 , λ1,......λm) Y = (m+1). For every z Z.
5. λ0 > 0.
6. f (x). . mi=1 λigi (x); that is f is linearly dependent on g1g2 .......gm.
Transcribed Image Text:
n kernel g/ f(x) eo 9(x)
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