HW1 - Math 175 Homework 1 Due Thursday, April 8 1. (a) Show...

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Math 175 Homework 1 Due Thursday, April 8 1. (a) Show that if a (real or complex) vector space V has an infinite dimensional subspace, then V is infinite dimensional. (b) Consider the normed linear space C ([0 , 1]) of real valued continuous func- tions on [0 , 1] with its usual norm (i.e. k f k = max [0 , 1] | f | ) and let S = span ± 1 ,x,x 2 ,... ² . Prove that S is not a closed subspace of C ([0 , 1]) , and hence in particular that S is infinite dimensional (hence C ([0 , 1]) is infinite dimensional by (a)). Notes: (a) V is finite dimensional if we can find a finite collection of vectors which span V . V is infinite dimensional if it is not finite dimensional. (b) Here (and subsequently) span { A } (with A a non-empty subset of a linear space X ) will mean the set of all possible linear combinations of finite collections of vectors taken from A . Thus, span { A } = n N i =1 λ i a i o , where λ i are scalars and a i A. (This is what your text calls lin ( A ) .) 2. Let l 2 R denote the set of all real sequences
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