Mathematics

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1500 questions
103
votes
5 answers

Space of bounded continuous functions is complete

I have lecture notes with the claim $(C_b(X), \|\cdot\|_\infty)$, the space of bounded continuous functions with the sup norm is complete. The lecturer then proved two things, (i) that $f(x) = \lim f_n (x)$ is bounded and (ii) that $\lim f_n \in…
Rudy the Reindeer
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103
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18 answers

In classical logic, why is $(p\Rightarrow q)$ True if $p$ is False and $q$ is True?

Provided we have this truth table where " $p\implies q$ " means "if $p$ then $q$ ": $$\begin{array}{|c|c|c|} \hline p&q&p\implies q\\ \hline T&T&T\\ T&F&F\\ F&T&T\\ F&F&T\\\hline \end{array}$$ My understanding is that " $p\implies q$ " means "when…
user701510
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103
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12 answers

Using "we have" in maths papers

I commonly want to use the phrase "we have" when writing mathematics, to mean something like "most readers will know this thing and I am about to use it". My primary question is whether this is too colloquial. My secondary question is what the…
Joe Tait
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103
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7 answers

Cardinality of set of real continuous functions

I believe that the set of all $\mathbb{R\to R}$ continuous functions is $\mathfrak c$ , the cardinality of the continuum . However, I read in the book "Metric spaces" by Ó Searcóid that set of all $[0, 1]\to\mathbb{R}$ continuous functions is…
kennytm
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103
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2 answers

Can we identify the duals of $\ell^\infty$ and $L^{\infty}$ with another "natural space"?

Can we identify the dual space of $l^\infty$ with another "natural space"? If the answer is yes, what can we say about $L^\infty$ ? By the dual space I mean the space of all continuous linear functionals.
omar
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103
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14 answers

What's the difference between $\mathbb{R}^2$ and the complex plane?

I haven't taken any complex analysis course yet, but now I have this question that relates to it. Let's have a look at a very simple example. Suppose $x,y$ and $z$ are the Cartesian coordinates and we have a function $z=f(x,y)=\cos(x)+\sin(y)$.…
Cancan
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103
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6 answers

Is the notorious $n^2 + n + 41$ prime generator the last of its type?

The polynomial $n^2+n+41$ famously takes prime values for all $0\le n\lt 40$. I have read that this is closely related to the fact that 163 is a Heegner number, although I don't understand the argument, except that the discriminant of $n^2+n+41$ is…
MJD
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103
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10 answers

Why is 'abuse of notation' tolerated?

I've personally tripped up on a few concepts that came down to an abuse of notation, and I've read of plenty more on stack exchange. It seems to all be forgiven with a wave of the hand. Why do we tolerate it at all? I understand if later on in one's…
Korgan Rivera
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103
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5 answers

If $x$, $y$, $x+y$, and $x-y$ are prime numbers, what is their sum?

Suppose that $x$, $y$, $x−y$, and $x+y$ are all positive prime numbers. What is the sum of the four numbers? Well, I just guessed some values and I got the answer. $x=5$, $y=2$, $x-y=3$, $x+y=7$. All the numbers are prime and the answer is $17$.…
chndn
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103
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4 answers

Why did no student correctly find a pair of $2\times 2$ matrices with the same determinant and trace that are not similar?

I gave the following problem to students: Two $n\times n$ matrices $A$ and $B$ are similar if there exists a nonsingular matrix $P$ such that $A=P^{-1}BP$. Prove that if $A$ and $B$ are two similar $n\times n$ matrices, then they have the same…
Taladris
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103
votes
19 answers

Unconventional mathematics books

I've recently purchased Oliver Byrne's reproduction of Euclid's Elements . It's a beautiful tome, that's rather unique in its presentation of the material as it demonstrates many of Euclid's proofs as lurid and lusciously coloured geometric figures.…
seeker
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102
votes
10 answers

Why can't you add apples and oranges, but you can multiply and divide them?

What is the algebraic difference between arithmetic operations, that prevents entities with different units from being summed or subtracted, but allows them to be multiplied or divided? This looks more like a question for Physics , but lengths and…
danza
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102
votes
28 answers

"Simple" beautiful math proof

Is there a "simple" mathematical proof that is fully understandable by a 1st year university student that impressed you because it is beautiful?
user138276
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102
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2 answers

What's the difference between simple induction and strong induction?

I just started to learn induction in my first year course. I'm having a difficult time grasping the concept. I believe I understand the basics but could someone summarize simple induction and strong induction and explain what the differences are?…
Jake Park
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102
votes
5 answers

What's going on with "compact implies sequentially compact"?

I've seen both counterexamples and proofs to "compact implies sequentially compact", and I'm not sure what's going on. Apparently there are compact spaces which are not sequentially compact; quick googling and wikipedia checks will turn up examples…
matt
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