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 …

May 12, 2024 · 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 …

What do finite, infinite, countable, not countable, countably infinite mean? [duplicate] Ask Question Asked 13 years, 2 months ago Modified 13 years, 2 months ago

Infinite decimals are introduced very loosely in secondary education and the subtleties are not always fully grasped until arriving at university. By the way, there is a group of very strict …

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 ∞ ∞ or infinite cardinals …

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.

Dec 3, 2020 · However, while Dedekind-infinite implies your notion even without the Axiom of Choice, your definition does not imply Dedekind-infinite if we do not have the Axiom of Choice …

Suppose there is a family (can be infinite) of measurable spaces. What are the usual ways to define a sigma algebra on their Cartesian product? There is one way in the context of defining …

Sep 5, 2021 · The reals can be thought of as a vector space over the rationals. The properties of a vector space are that addition and "scaling" by some scalar are well defined and …

I would like to have some examples of infinite dimensional vector spaces that help me to break my habit of thinking of $\\mathbb{R}^n$ when thinking about vector spaces.