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  3. 4 let v be a vector space recall that if...

Question: 4 let v be a vector space recall that if...

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4- Let V be a vector space. Recall that if a norm l ll on V is induced by an inner product on V, then for all f and g in V f-82 211f+2112 So if a norm on a vector space V does not satisfy the above identity, then that norm is NOT induced by an inner product. Use this fact to show that the o-norm on C(l-1,1) is not induced by an inner product. Hint: You may consider, for example, the functions f(x) and g(x)-*2 Remark. The same strategy can be used to show that if pf 2, then the p-norm on C([a, b) is NOT induced by an inner product. . Note that norm is a natural choice o norm on C ļa b , intuitively, as we discussed in class, the notion of distance obtained by o-norm measures the biggest gap between the graphs of two functions. However, as a consequence of what you proved in this exercise, the nice things we learned in class about approximation theory in inner product spaces cannot be directly applied to the case where CỦa, b) is equipped with the oo-norm.

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