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.
I am a little confused about how a cyclic group can be infinite. To provide an example, look at $\langle 1\rangle$ under the binary operation of addition. You can never make any negative numbers with
2 Infinite numbers do exist in the hyperreal number system which properly extends the real number system, but then their reciprocals are infinitesimals rather than zero. Thus the idea of $\frac {1} {0}$ can be interpreted as saying that if $\epsilon$ is infinitesimal then $\frac {1} {\epsilon}$ is infinite.
Before what follows, Cantor's diagonal argument was presented as a proof that $\mathbb {R}$ is uncountably infinite; this proof I found to be logically sound. However, after that, an alternative pro...
If the vector space is finite dimensional, then it is a countable set; but there are infinite-dimensional vector spaces over $\mathbb {Q}$ that are countable as sets.
By using infinite series, Newton was able to do a darn good description of gravitation, without pretending that his mathematical theory was the ultimate truth about the physical world (he knew about some deep problems connected with his theory, which couldn't be “explained” easily, such as instantaneous interaction between bodies far away ...
Can a countable set contain uncountably many infinite subsets such that the intersection of any two such distinct subsets is finite?
Can a countable set contain uncountably many infinite subsets such that ...