Let F([0, 1]) denote the vector space of all functions from the interval [0, 1] to R. (a) Let C([0, 1]) be the set of all continuous functions from [0, 1] to R. Show that C([0, 1]) is a vector subspace of F([0, 1]). (b) Let S be the set of all continuous functions from [0, 1] to R with f(0) = 0. Show that S is a vector subspace of C([0, 1]). (c) Let T be the set of all continuous functions from [0, 1] to R with f(0) = 1. Show that T is not a vector subspace of C([0, 1]).

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