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- Finding a basis of an infinite-dimensional vector space?
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
- infinity - What is the definition of an infinite sequence . . .
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?
- Proof of infinite monkey theorem. - Mathematics Stack Exchange
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
- Multiplication of infinite series - Mathematics Stack Exchange
Multiplication of infinite series Ask Question Asked 11 years, 9 months ago Modified 4 years, 9 months ago
- general topology - Why is the infinite sphere contractible . . .
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
- Infinite Cartesian product of countable sets is uncountable
So by contradiction, infinite $0-1$ strings are uncountable Can I use the fact that $\ {0,1\}$ is a subset of any sequence of countable sets $\ {E_n\}_ {n\in\mathbb {N}}$ and say the infinite product of this is uncountable too?
- If $S$ is an infinite $\sigma$ algebra on $X$ then $S$ is not countable
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
- How can Cyclic groups be infinite - Mathematics Stack Exchange
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
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