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 …

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...

Are you familiar with Taylor series? Series solutions of differential equations at regular points? From what foundation/background are you approaching this problem?

Nov 17, 2012 · The result is quite counter-intuitive. How can summing up products of finite numbers (the values of the random variable) with finite numbers (the probability of the random …

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.

Meaning of Infinite Union/Intersection of sets Ask Question Asked 8 years, 7 months ago Modified 4 years ago

Aug 4, 2016 · 0 Since singletons in R are closed in usual topology. We can think about infinite class of singletons {x} where x belongs to (0,1] then there union will be (0,1] which is not …

Oct 31, 2017 · Hence a vector space is infinite dimensional iff it has a basis which is infinite. If your infinite dimensional space has an inner product and is complete with respect to the …

Jun 6, 2020 · The reason being, especially in the non-standard analysis case, that "infinite number" is sort of awkward and can make people think about $\infty$ or infinite cardinals …

Mar 15, 2021 · I had a discussion with a friend about the monkey infinite theorem, the theorem says that a monkey typing randomly on a keyboard will almost surely produce any given books …