Most Popular
1500 questions
65
votes
9 answers
Linear Algebra Versus Functional Analysis
As it is mentioned in the answer by Sheldon Axler in this post , we usually restrict linear algebra to finite dimensional linear spaces and study the infinite dimensional ones in functional analysis. I am wondering that what are those parts of the…
Hosein Rahnama
- 15,676
65
votes
7 answers
Why is the Continuum Hypothesis (not) true?
I'm making my way through Thomas W Hungerfords's seminal text "Abstract Algebra 2nd Edition w/ Sets, Logics and Categories" where he makes the statement that the Continuum Hypothesis ( There does not exist a set with a cardinality less than the…
zetavolt
- 883
65
votes
12 answers
If a coin toss is observed to come up as heads many times, does that affect the probability of the next toss?
A two-sided coin has just been minted with two different sides (heads and tails). It has never been flipped before. Basic understanding of probability suggests that the probability of flipping heads is .5 and tails is .5. Unexpectedly, you flip the…
skurwa
- 787
65
votes
3 answers
Does multiplying all a number's roots together give a product of infinity?
This is a recreational mathematics question that I thought up, and I can't see if the answer has been addressed either. Take a positive, real number greater than 1, and multiply all its roots together. The square root, multiplied by the cube root,…
glowing-fish
- 623
65
votes
3 answers
Motivation to understand double dual space
I am helping my brother with Linear Algebra. I am not able to motivate him to understand what double dual space is. Is there a nice way of explaining the concept? Thanks for your advices, examples and theories.
Theorem
- 8,459
65
votes
6 answers
What does strength refer to in mathematics?
My professors are always saying, "This theorem is strong" or "There is a way to make a much stronger version of this result" or things like that. In my mind, a strong theorem is able to tell you a lot of important information about something, but…
Zachary F
- 1,922
65
votes
3 answers
Algebraic Topology Challenge: Homology of an Infinite Wedge of Spheres
So the following comes to me from an old algebraic topology final that got the best of me. I wasn't able to prove it due to a lack of technical confidence, and my topology has only deteriorated since then. But, I'm hoping maybe someone can figure…
squiggles
- 1,953
65
votes
15 answers
Is there any geometric way to characterize $e$?
Let me explain it better: after this question, I've been looking for a way to put famous constants in the real line in a geometrical way -- just for fun. Putting $\sqrt2$ is really easy: constructing a $45^\circ$ - $90^\circ$ - $45^\circ$ triangle…
Ian Mateus
- 7,601
65
votes
4 answers
Proving $\frac{\sin x}{x} =\left(1-\frac{x^2}{\pi^2}\right)\left(1-\frac{x^2}{2^2\pi^2}\right) \left(1-\frac{x^2}{3^2\pi^2}\right)\cdots$
How to prove the following product? $$\frac{\sin(x)}{x}= \left(1+\frac{x}{\pi}\right) \left(1-\frac{x}{\pi}\right) \left(1+\frac{x}{2\pi}\right) \left(1-\frac{x}{2\pi}\right) \left(1+\frac{x}{3\pi}\right) \left(1-\frac{x}{3\pi}\right)\cdots$$
Michael Li
- 2,291
65
votes
6 answers
If we randomly select 25 integers between 1 and 100, how many consecutive integers should we expect?
Question: Suppose we have one hundred seats, numbered 1 through 100. We randomly select 25 of these seats. What is the expected number of selected pairs of seats that are consecutive? (To clarify: we would count two consecutive selected seats as a…
David Steinberg
- 1,479
65
votes
8 answers
What is Cauchy Schwarz in 8th grade terms?
I'm an 8th grader. After browsing aops.com, a math contest website, I've seen a lot of problems solved by Cauchy Schwarz. I'm only in geometry (have not started learning trigonometry yet). So can anyone explain Cauchy Schwarz in layman's terms, as…
Schneider
- 749
65
votes
3 answers
Is $\lfloor n!/e\rfloor$ always even for $n\in\mathbb N$?
I checked several thousand natural numbers and observed that $\lfloor n!/e\rfloor$ seems to always be an even number. Is it indeed true for all $n\in\mathbb N$? How can we prove it? Are there any positive irrational numbers $a\ne e$ such that…
july
- 1
65
votes
1 answer
Does $\Bbb{CP}^{2n} \mathbin{\#} \Bbb{CP}^{2n}$ ever support an almost complex structure?
This question has now been crossposted to MathOverflow, in the hopes that it reaches a larger audience there. $\Bbb{CP}^{2n+1} \mathbin{\#} \Bbb{CP}^{2n+1}$ supports a complex structure: $\Bbb{CP}^{2n+1}$ has an orientation-reversing diffeomorphism…
user98602
65
votes
3 answers
What is difference between a ring and a field?
The ring axioms require that addition is commutative, addition and multiplication are associative, multiplication distributes over addition. A field can be thought of as two groups with extra distributivity law. A ring is more complex: with abelian…
zinking
- 1,137
65
votes
1 answer
Lebesgue measurable set that is not a Borel measurable set
exact duplicate of Lebesgue measurable but not Borel measurable BUT! can you please translate Miguel's answer and expand it with a formal proof? I'm totally stuck... In short: Is there a Lebesgue measurable set that is not Borel measurable? They are…
example
- 2,185