What is Functional Analysis and How Can It Be Applied in Science?

math8
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I really don't think there is much to show. How do you define an infinite sum like the sum of x_k*e_k? I would say it's the sequence whose i-th term is the sum of the i-ith terms of all of the x_k*e_k. So for a given i there's only one sequence with a nonzero term. You definitely don't want to start trying to prove the partial sums converge in the l_infinity norm. They don't unless x converges to zero (in the real infinite sequence sense).
 
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There are two things I don't understand about this problem. First, when finding the nth root of a number, there should in theory be n solutions. However, the formula produces n+1 roots. Here is how. The first root is simply ##\left(r\right)^{\left(\frac{1}{n}\right)}##. Then you multiply this first root by n additional expressions given by the formula, as you go through k=0,1,...n-1. So you end up with n+1 roots, which cannot be correct. Let me illustrate what I mean. For this...
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