In C 0 1 Let F T T And G T Et Compute F G As Defined In Example 3 F G And F 9 Then Verify 1 (229.42 KiB) Viewed 21 times
= In C([0,1]), let f(t) = t and g(t) = et. Compute (f,g) (as defined in Example 3), ||f|| ||g||, and f +9||. Then verify both Cauchy's inequality and the triangle inequality.
= Example 3 Let V C([0, 1]), the vector space of real-valued continuous functions on [0, 1]. For f, g € V, define (f,g) = So f(t)g(t) dt. Since the preceding integral is linear in f, (a) and (b) are immediate, and (c) is trivial. If f + 0, then f2 is bounded away from zero on some subinterval of [0, 1] (continuity is used here), and hence (f. f) = S LF(t))? dt > 0.
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