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[QUOTE="Office_Shredder, post: 6827430, member: 53426"] Here's a brief outline of an idea. I might have time to flesh it out this weekend, or maybe an enterprising student who wants to take a crack at this but isn't sure where to start is inspired to go through in detail and see if it turns into a proof. [Spoiler] since terns of the sum go to zero, you know that ##x_n\approx y_n## for large ##n##. So there's some ##N## for which every element of ##\span(x_{N+1},...)## is close to an element of ##\span(y_{N+1},...)## and vice versa. This might require using the convergence of the sum, in some additional way. ##x_1,...,x_N## and ##y_1,...,y_N## span ##N## dimensional spaces ##X## and ##Y##. Every element of ##Y## is in the span of all the x's and is almost orthogonal to the infinite dimensional space in the last paragraph, so is well approximated by an element in ##X##. Since ##X## and ##Y## have the same dimension, every element of ##X## is well approximated by an element of ##Y## as well. If the y's do not span the space, there is some vector orthogonal to them. But that vector cannot be both almost orthogonal to ##X## and also almost orthogonal to the span of the other x's, which means it cannot be orthogonal to both ##Y## and the span of the other y's. [/spoiler] [/QUOTE]
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Mathematics
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Orthonormal Bases on Hilbert Spaces
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