The infinite manifold of two or three dimensions, the mathematical beings which depend on a number of variables greater even than three, any number in fact, still have no greater power than the linear …

Oct 4, 2020 · 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 …

Dec 10, 2023 · Here's a proof by induction that the resistance of a finite version of this ladder with $\ n\ $ rungs is indeed homogeneous of degree $1$ in the variable $\ R\ .$ Taking the limit as $\ …

An infinite number? Kind of, because I can keep going around infinitely. However, I never actually give away that sweet. This is why people say that 1 / 0 "tends to" infinity - we can't really use infinity as a …

Write out a few terms of the series. You should see a pattern! But first consider the finite series: $$\sum\limits_ {n=1}^ {m}\left (\frac {1} {n}-\frac {1} {n+1 ...

Jan 31, 2025 · There are the following textbooks to learn about infinite-dimensional manifolds: "The Convenient Setting of Global Analysis" by Andreas Kriegl and Peter W. Michor

Feb 20, 2019 · The user @Eevee Trainer provided a nice explanation on how we define infinite nested radical in terms of limit of finite nested radical which should be insensitive of the starting point.

The dual space of an infinite-dimensional vector space is always strictly larger than the original space, so no to both questions. This was discussed on MO but I can't find the thread.

Nov 7, 2022 · I see what you mean, so does a normed-space being infinite means that it maps a vector space to a continous interval? If this is the case how do we have a finite normed-space?

Dec 30, 2019 · However, there is also a theorem that states that every vector space (finite- or infinite-dimensional) has a basis. So my question is how a basis can even exist for the infinite-dimensional …