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For many infinite-dimensional vector spaces of interest we don't care about describing a basis anyway; they often come with a topology and we can therefore get a lot out of studying dense subspaces, some of which, again, have easily describable bases
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The infinite monkey theorem states that if you have an infinite number of monkeys each hitting keys at random on typewriter keyboards then, with probability 1, one of them will type the complete works of William Shakespeare
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Why is the infinite sphere contractible? I know a proof from Hatcher p 88, but I don't understand how this is possible I really understand the statement and the proof, but in my imagination this
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6 Show that if a $\sigma$-algebra is infinite, that it contains a countably infinite collection of disjoint subsets An immediate consequence is that the $\sigma$-algebra is uncountable
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Multiplication of infinite series Ask Question Asked 11 years, 9 months ago Modified 4 years, 9 months ago
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Except for $0$ every element in this sequence has both a next and previous element However, we have an infinite amount of elements between $0$ and $\omega$, which makes it different from a classical infinite sequence So what exactly makes an infinite sequence an infinite sequence? Are the examples I gave even infinite sequences?
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Does infinite equal infinite? Ask Question Asked 11 years, 10 months ago Modified 5 years, 1 month ago
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Are you familiar with Taylor series? Series solutions of differential equations at regular points? From what foundation background are you approaching this problem?
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