Analysis II | Part IB, 2003

Let VV be the vector space of continuous real-valued functions on [1,1][-1,1]. Show that the function

f=11f(x)dx\|f\|=\int_{-1}^{1}|f(x)| d x

defines a norm on VV.

Let fn(x)=xnf_{n}(x)=x^{n}. Show that (fn)\left(f_{n}\right) is a Cauchy sequence in VV. Is (fn)\left(f_{n}\right) convergent? Justify your answer.

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