4.I 3 B

Analysis II | Part IB, 2005

Let VV be the vector space of continuous real-valued functions on [0,1][0,1]. Show that the function

f=01f(x)dx\|f\|=\int_{0}^{1}|f(x)| d x

defines a norm on VV.

For n=1,2,n=1,2, \ldots, let fn(x)=enxf_{n}(x)=e^{-n x}. Is fnf_{n} a convergent sequence in the space VV with this norm? Justify your answer.

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