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Q.IL. Let X be a space of real-valued polynomials of one variable x. For f(x). g(x) e X, define the inner product ((x).g(x) by the for- mula (f(x),g(x))-匠.e-rf(x)g(x)dx and the norm Ilf(x)1|- (ro.so) (o) By using the equality d- Vr and integrating by parts, obain th inN. ou fin it e dx for all n e N. (If you find it difficult, do for n 1,2,3.) and fi(x) - cie satisfies the equality foe) (a(x). fo(x) -0 nomial fs(x) c3 (2) Choose positive numbers co. cı so that the polynomial fo(x) - co lA) 1, (3) Choose a positive number cz and a real number c2 so that the poly- nomial f2(x) cx2+ satisfies the equalities |l(0)l- 1 and (4) Choose a positive number cs and a real number c so that the poly- tiss he equalitiesl- 1 and +ct satisfies the equalities ls(x) -1 and (fs(x).fi(x)) 0
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