Tagged Questions

Examples and counterexamples are great ways to learn about the intricacies of definitions in mathematics. Counterexamples are especially useful in topology and analysis where most things are fairly intuitive, but every now and then one may run into borderline cases where the naive intuition may ...

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2
votes
2answers
555 views

How do you prove the $p$-norm is not a norm in $\mathbb R^n$ when $0<p<1$?

I see that it fails to satisfy the triangle inequality by example but I don't see how to prove this is the case for all $0 < p < 1$. The definition I am using for $p$-norm is $$ \|A\|_p= ...
4
votes
1answer
469 views

Spaces with equal homotopy groups but different homology groups?

Since it's fairly easy to come up with a two spaces that have different homotopy groups but the same homology groups ($S^2\times S^4$ and $\mathbb{C}\textrm{P}^3$). Are there any nice examples of ...
20
votes
3answers
3k views

Discontinuous linear functional

I'm trying to find a discontinuous linear functional into $\mathbb{R}$ as a prep question for a test. I know that I need an infinite-dimensional Vector Space. Since $\ell_2$ is infinite-dimensional, ...
2
votes
1answer
178 views

Differentiability of a function, of its sections and of its components

Let's consider two cases: For $f:\mathbb{R}^2 \to \mathbb{R}$, where the domain may be some open subset in $\mathbb{R}^2$, define its sections to be functions in $\{ f(,x_2), f(x_1,), \forall ...
2
votes
5answers
5k views

List applications of sets & relations in science/business/tech that a highschooler can understand

What are some applications of sets & relations in science/business/tech that a highschooler can understand? To kindle a young mind, what examples can be given?
3
votes
1answer
888 views

graphs of functions which are closed, but fail to be continuous

I tried looking up a question regarding graphs of continuous functions on this site, but all the ones I found consider functions from $\mathbb{R}$ into $\mathbb{R}$. I have been pondering the ...
3
votes
3answers
608 views

Looking for examples of first countable, compact spaces which is not separable

Could someone give me some classical examples of first countable, compact spaces which is not separable? However, other examples are also welcome. Any help will be appreciated.
4
votes
1answer
108 views

If $M_0$, $M_1$, and $M_2$ are least upper bounds of $|f(x)|$, $|f'(x)|$ and $|f''(x)|$, does $M_1^2\leq 4M_0M_2$ for vector valued functions?

If $a\in\mathbb{R}$, $f\colon(a,\infty)\to\mathbb{R}$ is a twice-differentiable function, and $M_0$, $M_1$, and $M_2$ are least upper bounds of $|f(x)|$, $|f'(x)|$ and $|f''(x)|$, then ...
0
votes
1answer
523 views

Relation between Fourier transform and Fourier series

Let $f$ be a function on $\mathbb R^n$ whose Fourier transform $\hat f$ exists. Is there any relation between integrability of $\hat f$ and summability of the series $\sum_{n \in \mathbb Z^n} \hat ...
5
votes
1answer
178 views

Nonamenable subgroups of the unitary group of the hyperfinite II_1 factor

The hyperfinite $II_1$ factor arises as the group von Neumann algebra of any infinite amenable group such that every conjugacy class but that of the identity has infinite cardinality. The unitary ...
13
votes
1answer
459 views

Complete space as a disjoint countable union of closed sets

It is a consequence of Baire's theorem that a connected, locally connected complete space cannot be written $$ X = \bigcup_{n \geq 1}\ F_n$$ where the $F_n$ are nonempty, pairwise disjoint closed ...
9
votes
4answers
2k views

Connected metric spaces with at least 2 points are uncountable.

That's a problem I proved (quite a while back) in tiny Rudin. However, I don't really get it. The other questions were actually useful results - I don't think I've ever come near using this result. ...
14
votes
3answers
533 views

Why does the Continuum Hypothesis make an ideal measure on $\mathbb R$ impossible?

On the page 43 of Real Analysis by H.L. Royden (1st Edition) we read: "(Ideally) we should like $m$ (the measure function) to have the following properties: $m(E)$ is defined for each subset $E$ of ...
3
votes
1answer
196 views

Dense pre-images implies continuous right inverse?

Suppose $f : \mathbb R \to \mathbb R$ is such that pre-image of every point under $f$ is dense in $\mathbb R$. This, of course, implies that $f$ is surjective, and hence has a right inverse ...
2
votes
1answer
207 views

Is there a topological space which is star compact but not star countable?

A topological space $X$ is said to be star compact if whenever $\mathscr{U}$ is an open cover of $X$, there is a compact subspace $K$ of $X$ such that $X = \operatorname{St}(K,\mathscr{U})$. A ...
1
vote
1answer
210 views

Is there a Tychonoff space with $G_\delta$ diagonal that isn't submetrizable?

Recently, I'm interested in the topological spaces with $G_\delta$ diagonal. Could someone give me some examples such that the given topology space is a Tychonoff space with a $G_\delta$ diagonal but ...
2
votes
0answers
49 views

examples of 'continuous bases of functions,' like the Fourier transform

For suitable choice of a one-parameter family of functions $\{ g_w:\mathbb{R} \rightarrow \mathbb{C} \}_{w\in \mathbb{R}}$, the following two statements are equivalent (modulo sets of measure $0$): ...
2
votes
1answer
172 views

Basic Properties of Convergence in distribution

Suppose real-valued random variables $X_n$ converge to $X$ in distribution. Suppose $x_n$ be a sequence of numbers that for each $n$, $\{x_n \}$ is in the support of the distribution of $X_n$. Suppose ...
31
votes
5answers
1k views

False beliefs about Lebesgue measure on $\mathbb{R}$

I'm trying to develop intuition about Lebesgue measure on $\mathbb{R}$ and I'd like to build a list of false beliefs about it, for example: every set is measurable, every set of measure zero is ...
4
votes
1answer
197 views

An example of a family of atomic measures whose sum is not atomic

I look for a example of family of atomic measures such that their sum is not atomic. A measure $\mu$ on a $\sigma$-algebra $S$ of subsets of $X$ is called atomic if every measurable set of positive ...
8
votes
2answers
419 views

Example of finitely generated subgroups whose intersection is not finitely generated

I'm reading G.Graetzer's Lattice Theory: First Concepts and Distributive Lattices and working on its exercises. One of them is to prove $(A, \subset)$, where $A$ is the set of finitely generated ...
11
votes
4answers
2k views

Famous uses of the inclusion-exclusion principle?

The standard textbook example of using the inclusion-exclusion principle is for solving the problem of derangement counting; using inclusion-exclusion (and some basic analysis) it can be shown that ...
5
votes
1answer
310 views

Non-Noetherian rings with an ideal not containing a product of prime ideals

It is well-known that in every commutative Noetherian ring every ideal contains a product of prime ideals. Are there examples of non-Noetherian rings with an ideal that does not contain any prime ...
7
votes
4answers
781 views

Two CW complexes with isomorphic homotopy groups and homology, yet not homotopy equivalent

A standard example of two CW complexes which have isomorphic homotopy groups but are not homotopy equivalent is $ RP^2 \times S^3$ and $RP^3 \times S^2$. The easiest way to see that they are not ...
2
votes
1answer
106 views

Element $a$ of a C*-algebra such that $f(a^*a) = 0$ but $f(aa^*) \neq 0$ for some positive linear functional $f$

Could somebody please give me an example, assuming one exists, of an element $a$ of a C*-algebra $A$ such that $f(a^*a) = 0$ but $f(aa^*) \neq 0$ for some positive linear functional $f:A \to ...
5
votes
1answer
1k views

Intersection of compact sets

I have a brief question about Theorem 2.36 in Baby Rudin. The theorem is as follows: If $\{K_\alpha\}$ is a collection of compact subsets of a metric space $X$ such that the intersection of every ...
4
votes
2answers
438 views

Left and right ideals of $R=\left\{\bigl(\begin{smallmatrix}a&b\\0&c \end{smallmatrix}\bigr) : a\in\mathbb Z, \ b,c\in\mathbb Q\right\}$

If $$R=\left\{ \begin{pmatrix} a &b\\ 0 & c \end{pmatrix} \ : \ a \in \mathbb{Z}, \ b,c \in \mathbb{Q}\right\} $$ under usual addition and multiplication, then what are the left and right ...
1
vote
2answers
167 views

If a sub-C*-algebra does not contain the unit, is it contained in a proper ideal?

Suppose that $R$ and $S$ are unital rings and that $S$ is a subring of $R$ in the weak sense where the multiplicative identities $1_R$ and $1_S$ are not assumed to be the same. In fact, assume $1_R ...
8
votes
1answer
648 views

Weird measurable set

In the following, consider the Lebegue measure in $\mathbb{R}^d$. Consider $E\subseteq \mathbb{R}^d$ measurable, with $0\lt m(E)\lt\infty$, such that any measurable subset $F$ of $E$ satisfies ...
7
votes
1answer
517 views

Sum of Cauchy Sequences Cauchy?

Let $(X,+)$ be an abelian group and $d$ a metric on $X$. Suppose $\{a_n\}$ and $\{b_n\}$ are Cauchy sequences. What conditions on the relation between the group operation and the metric are sufficient ...
1
vote
2answers
350 views

Measure Product Theorem: may non-$\sigma$-finiteness result unique product?

Let $i\in\{1,2\}$. The Measure Product Theorem states that, given the measure spaces $(X_i,\Sigma_i,\mu_i)$, there is at least one product measure $\pi$ such that $\pi(A_1\times ...
5
votes
2answers
10k views

Examples of Simpson's Paradox

I'm looking for fresh examples of Simpson's paradox for use in my statistics courses. The examples I've been using are fine, but I'd like to have some new ones, and I'm hoping folks here might know a ...
2
votes
2answers
61 views

if $ax|n$ and $ax+1$ is prime does $ax+1|a^{n}-1$?

Are there any $a,x,n$ such that $ax|n$ and $ax+1$ is prime but $a^{n}-1$ is not a multiple of $ax+1$, apart from $a=x=n=1$? I had an answer to a related question earlier: Can $x^{n}-1$ be prime if ...
28
votes
5answers
2k views

An example of an easy to understand undecidable problem

I am looking for an undecidable problem that I could give as an easy example in a presentation to the general public. I mean easy in the sense that the mathematics behind it can be described, well, ...
6
votes
2answers
736 views

Opposite of a contraction mapping

I am taking Real Analysis and we recently went over the Banach Fixed-point Theorem, also commonly known as the Contraction Mapping Theorem which states: If $(X,d)$ is a complete metric space, and ...
0
votes
2answers
200 views

Can someone give me a counterexample to disprove this statement?

Claim : For any even number $n$ there is at least one prime number of the form : $$p=k\cdot2^{n}-1$$ with following properties : $k=2^{a-n}+1 , n\leq a < 2n , $ and $a,n\in ...
4
votes
1answer
122 views

Bidual of a WSC space

Let $E$ be a Banach space which is weakly sequentially complete (i.e. each weak Cauchy sequence converges weakly). Must $E^{**}$ be weakly sequentially complete either? Of course, this question is ...
2
votes
1answer
190 views

Give an example that $\overline{A \cap B} \neq \overline{A} \cap \overline{B}$ [duplicate]

Possible Duplicate: Is the closure of $ X \cap Y$ equal to $\bar{X} \cap \bar{Y}$? I'm sorry to ask another question so soon after my last one, but my exam Introduction to Functional ...
3
votes
2answers
972 views

Contraction mapping does not hold in metric space

Let $X=\mathbb{Q}\cap [1,2]$, i.e $X$ is the set of rational number between 1 and 2 inclusive. We can consider $X$ to be a metric space by endowing it with the usual distance function, i.e for $x,y ...
4
votes
1answer
667 views

Are all continuous everywhere but nowhere differentiable functions fractal in nature?

Functions like the Weierstrass function or van der Waerden's function exhibit self-similar plots. Is this characteristic of continuous everywhere, differentiable nowhere functions? Is there a ...
12
votes
5answers
2k views

A compact Hausdorff space that is not metrizable

Is there an example of a compact Hausdorff space that is not metrizable? I was thinking maybe the space of continuous functions $f: X \rightarrow Y$ between topological spaces $X, Y$, might work, but ...
1
vote
0answers
213 views

Is there a ccc but not separable space $X$ with a zeroset-diagonal, that isn't submetrizable?

Is there a ccc but not separable space $X$ with a zeroset-diagonal, that isn't submetrizable? separable = $X$ has a countable dense subset. A space $X$ has a zeroset-diagonal when there is a ...
9
votes
3answers
477 views

Is there an easy example of a vector space which can not be endowed with the structure of a Banach space

Let $V$ be a real vector space. Is there always a norm on $V$ such that $V$ is complete with respect to this norm? If not, is there an easy counterexample?
5
votes
2answers
343 views

Famous Finite Sets [closed]

What are the most famous (or most beautiful, IYO) finite sets in mathematics? I'm especially looking for 'large' sets that contain more than $2^{10} \approx 1000$ but fewer than $2^{20} \approx ...
4
votes
3answers
435 views

Is total boundedness a topological property?

If a metrizable topological space is totally bounded with one metric, is it totally bounded with all others? A related, stronger question: if every metrization of a topological space is bounded, are ...
2
votes
1answer
135 views

Multiplier of periodic orbits

I have $z\mapsto 1-1/z^2$ which has the periodic orbit {$1,0,\infty$} on the Riemann sphere. Next, I want to calculate the corresponding multiplier $\lambda= (f^{\circ n})' (z_i)=f'(z_1)\cdots ...
4
votes
1answer
263 views

An explicit example of a differentiable function taking rational values at rational points but whose derivative is irrational at rational points

Construct an example of a differentiable function such that $$ \forall r \in {\Bbb Q}\quad f(r) \in {\Bbb Q}\text{ but } f'(r) \notin {\Bbb Q} $$ this example is not trivial, in a paper they ...
13
votes
1answer
235 views

Measurable subset of $\mathbb{R}$ with a specific property

Let $A$ be a subset of $\mathbb{R}$ such that its intersection with every finite segment is Lebesgue measurable. I am looking for an example of such an $A$ with the additional property that the ...
15
votes
1answer
967 views

What does “locally trivial” do for us?

For the following we will work in the smooth category. (But examples in the topological category is also welcome.) The usual definition of a fibre bundle is Def A fibre bundle is the quadruple ...
2
votes
2answers
259 views

Dense subset of the plane that intersects every rational line at precisely one point?

It seems there should exist a non-measurable bijection $f: \mathbb{R}\rightarrow \mathbb{R}$. And thus we can obtain a non-measurable graph on $\mathbb{R}^2$ which intersects every horizontal or ...