Recall that we considered the vector space of real-valued, continuous functions on [0, 1], de- noted...

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Recall that we considered the vector space of real-valued, continuous functions on [0, 1], de- noted by C[0, 1]. Equip this space with the standard inner product, with (f, g) = f f(t)g(t)dt. Also recall the sequence of functions (fn)=1 defined as follows 0 if 0<t</ fn(t)=2nt + (1-n) if //-// <t</ if < t < 1. - {ame v Show that for any finite n, fn is a continuous function. (b) Now consider any pair of positive integers n < m, and evaluate the norm ||fn - fm. Your expression should be explicit and depend on n and m (recall that the norm is the one induced by the inner product). (c) Using your expression above, show that the sequence (fn) is a Cauchy sequence. Conclude that the inner product space introduced above is not complete by thinking about foo. Recall that we considered the vector space of real-valued, continuous functions on [0, 1], de- noted by C[0, 1]. Equip this space with the standard inner product, with (f, g) = f f(t)g(t)dt. Also recall the sequence of functions (fn)=1 defined as follows 0 if 0<t</ fn(t)=2nt + (1-n) if //-// <t</ if < t < 1. - {ame v Show that for any finite n, fn is a continuous function. (b) Now consider any pair of positive integers n < m, and evaluate the norm ||fn - fm. Your expression should be explicit and depend on n and m (recall that the norm is the one induced by the inner product). (c) Using your expression above, show that the sequence (fn) is a Cauchy sequence. Conclude that the inner product space introduced above is not complete by thinking about foo.