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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What would be an example such that $\langle a\rangle\langle b\rangle \neq \langle ab\rangle$?

Let $R$ be an rng (no unity). Define $IJ$ as the ideal generated by $\{ab:a\in I, b\in J\}$ for every ideals $I,J$ of $R$. Let $I=\langle a\rangle , J=\langle b\rangle $ be principal ideals. What ...
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27 views

Existence Adjoint Operator: Boundedness?

Context This would make the check on the GNS construction much more simple. Problem Given a Hilbert space $\mathcal{H}$. Consider a merely linear operator $A:\mathcal{H}\to\mathcal{H}$. Suppose ...
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1answer
42 views

Examples of Free Lie Algebra

In wikipedia, free Lie algebras are defined using the universal property. Can anyone give some concrete examples of free Lie algebras? "In mathematics, a free Lie algebra, over a given field $K$, is ...
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1answer
58 views

Prove or disprove regarding sequences

Question: Find a sequence $\displaystyle \{a_n\}_{n=1}^{\infty}$ such that $a_n\rightarrow 0$ and $n\left|a_{n+1}-a_n\right|\rightarrow \infty$. If no such sequence exists, prove it. My try: At ...
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1answer
23 views

Finding an specific counterexample of the interchanging of the limit and integral.

Hi I found the following exercise where I'm stuck, I'd appreciate any help. Thanks in advances Let $0\le f_n\to f$ and $\int f_n\to c>0$ pointwise. Show that $\int \lim f_n d\mu\in [0,c]$ and ...
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2answers
55 views

Fun results from modular arithmetic

I'm trying to go through various bits of neat, fun math with some junior-high-school students in my local area, and am thinking of doing a short unit on modular arithmetic/finite groups. I'm looking ...
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0answers
81 views

Connected space such that (almost) no subspace is connected [duplicate]

Is there a connected space $(X,\tau)$ such that $X$ has more than $2$ points and the only proper connected subsets of $X$ are the singletons?
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1answer
182 views

Can the composite of two projections really fail to be a projection?

Let $H$ denote a Hilbert space. For any closed subspace $C \subseteq H$, write $P_C$ for the orthogonal projection onto $C$. Then according to wikipedia, the composite $P_U \circ P_V$ needn't be a ...
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2answers
64 views

Zero mean but not a martingale

I am looking for a simple stochastic process which has zero mean for all $t\geq0$ but it is not a martingale. I been looking in to local martingales but having trouble keeping the mean zero for all ...
0
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1answer
23 views

What is an example of ideals such that $IJ\neq JI$?

Let $R$ be an rng. Let $I,J$ be ideals of $R$. What is an example of $IJ\neq JI$?
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1answer
83 views

Counter-example for abelian category that is not concrete

I am trying to figure out a counter-example for abelian category that is not concrete category. Does the category of representations over a quiver work? Thanks,
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3answers
40 views

Interior of a set?

I'm trying to think if their is any topology for which this is false: If G is an open set, then G = interior(G) Can anybody think of anything? I'm pretty sure it's straight forward.
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1answer
95 views

Prove using the smallest counterexample technique that: $\binom {2n}n\leqslant4^n.$

Actually what i know is that i must assume that this statement is false and then try to come up with non sense statement. Prove by the smallest counterexample technique the statement $$\binom ...
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1answer
49 views

A polynomial algebra that is free as an $A$-module

I'm working through some problems when I stumbled across a question asking about conditions for when the polynomial algebra $k[x_1,\ldots,x_n]$ is also a free $A$-module, where $A$ is some ...
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5answers
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Humorous integration example?

I was just reading though an introductory calculus book and it has the note: NOTE When integrating quotients, do not integrate the numerator and denominator separately. This is no more valid in ...
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1answer
140 views

Topological spaces with unknown fundamental group

Are there any well known topological spaces for which the fundamental group is not known yet?
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3answers
54 views

An example of a Banach space whose evaluation map is not surjective?

I have been giving the following corollary while studying functional analysis Let $X$ be a normed vector space. Then the evaluation map $$ev : X \to X'' , x \mapsto (f \mapsto fx) $$ is an ...
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2answers
43 views

Hilbert spaces of holomorphic functions

Could you please give me some examples of Hilbert spaces of holomorphic functions? Or even books or notes on Hilbert spaces of holomorphic functions? I need just a good number of examples and perhaps ...
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1answer
30 views

Examples of the Geometric Realization of a Semi-Simplicial Complex

I am reading The Geometric Realization of a Semi-Simplicial Complex by J. Milnor and here are the definitions: I find it difficult to visualize without specific examples. Can anyone help to provide ...
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1answer
89 views

A counterexample to $Ae\simeq Ae^{\prime}\to A(1-e)\simeq A(1-e^{\prime})$

Let $e,e^{\prime}$ be two idempotents in a $k$- algebra $A$ ($k$ is a field) . Then my guess $Ae\simeq Ae^{\prime}$ (as a left $A$- module) does not imply $A(1-e)\simeq A(1-e^{\prime})$ in general, ...
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2answers
39 views

A function with a finite limit at (both) 0 and infinity

I ask myself if there exists a $f(x)$ function that limit goes to a finite number for both when x goes to 0 and to infinity. Is it possible in some way ?
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2answers
75 views

Give a counterexample to show that $(AB)^{-1} \neq A^{-1}B^{-1}$

Give a counterexample to show that $(AB)^{-1}$ doesn't equal $A^{-1}B^{-1}$ I'm not sure how to approach this, so I just used the idea that the matrix multiplication is not commutative. so it goes: ...
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3answers
54 views

$f_n \to 0$ $ a.e.$ and $\lim \int f_n d\mu =0$ but $\sup_n f_n$ is not in $L^1$

Give an example of a finite measure space $(X,M,\mu)$ and a sequence of functions $f_n:X \to[0, \infty)$ such that $f_n \to 0$ $a.e.$ and $\lim \int f_n d\mu=0$ but $\sup_n f_n$ is not in $L^1$ I ...
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1answer
21 views

Show f need not be continuous.

If the functions $f_k$ are lower semicontinuous, $f_k \rightarrow f$ pointwise, and $f_{k+1} \geq f_k(x)$ then $f$ is lower semicontinuous. Show that $f$ need not be continuous even if the $f_k$ are ...
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0answers
41 views

Find counter-example about product measure

This problem is in Real analysis for graduate students, Richard F. Bass. Problem is the following Let $(X,\mathcal A)$, $(Y,\mathcal B)$ be two measurable spaces and let $f \ge 0 $ be measurable ...
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Examples of types of mathematical models

I am a student currently doing a course on modelling and simulation. I came across the classifications of mathematical models and studied that they can classified as static or dynamic, deterministic ...
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1answer
33 views

Is $N_G(S)=N_G(\langle S\rangle)$?

Related question: What would be a counterexample of $N_G(T)\not\subset N_G(S)$? Let $G$ be a group. Let $S$ be a subset of $G$. Then, is $N_G(S)=N_G(\langle S\rangle)$? I have proved that ...
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1answer
58 views

Is a topological space determined by its components and their quotient?

Given connected topological spaces $X_i$ and a totally disconnected space $Y$, is there a unique topological space $X$ with components homeomorphic to $X_i$ and $X/\sim$ homeomorphic to $Y$? ($\sim$ ...
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1answer
24 views

What would be a counterexample of $N_G(T)\not\subset N_G(S)$?

Let $G$ be a group and $S,T$ be subgroups of $G$ such that $S\subset T$. Is there an example such that $N_G(T)\not\subset N_G(S)$? Also, what is an example such that $N_G(S)\not\subset N_G(T)$?
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463 views

Some examples of virtually cyclic groups

The only virtually cyclic groups (ie. groups containing $\mathbb{Z}$ as subgroup of finite index) I really know are : the groups $F \times \mathbb{Z}$, where $F$ is a finite group, and the infinite ...
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1answer
53 views

Nowhere differentiability of Weierstrass function

It's again from Tao's book. Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be the function $$f:= \sum_{n=1}^\infty 4^{-n} \sin(8^n\pi x)$$ Show that for every 8-dyadic interval ...
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2answers
53 views

Nested Sum Encountered in Maclaurin Expansion of $e^{-x^2}$

In the course of working out the Maclaurin expansions of $e^{-x^2}$ and $cos(x^2)$, I ran into the following nested sum: $$ \underbrace{ \sum_{a=0}^1 \left( a \sum_{b=0}^{a+1} b \left( ...
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1answer
124 views

An abelian Banach algebra without characters

Can one give an example of an abelian Banach algebra with empty character space? Such algebra must be necessarily non-unital. I couldn't find any examples of such algebras. Thanks!
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31answers
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What are some conceptualizations that work in mathematics but are not strictly true?

I'm having an argument with someone who thinks it's never justified to teach something that's not strictly correct. I disagree: often, the pedagogically most efficient way to make progress is to ...
7
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3answers
244 views

What's a good motivating example for the concept of a slice category?

What nice example can one give a beginner to really motivate the idea of a slice category, before they've met the more general notion of a comma category? There's the toy example of a poset category ...
2
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1answer
25 views

Does such a finite monoid exist?

Does there exist a finite monoid $M$ such that for some $x \in M,$ the following hold? $x$ cancels on both the left and the right: $$\frac{ax=bx}{a=b}\qquad \frac{xa=xb}{a=b}$$ $x$ has no two-sided ...
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2answers
29 views

Example of a normal extension.

Can you give an example of a Normal extension which is not a splitting field of some polynomial.? I know that splitting field of a polynomial is always a normal extension but i am looking for the ...
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3answers
206 views

Prove or disprove that if $f$ is continuous function and $A$ is closed, then $\,f[A]$ is closed.

If $f:X\to Y$ is a continuous function and $A\subset X$ is a closed set, then $f[A]$ is also closed. I know that if $f[A]$ is closed implies $A$ is closed then $f$ is continuous, but I'm not sure ...
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2answers
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1
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2answers
60 views

Finding a function $\mathbb{N} \to \mathbb{N}$ that is surjective but not injective.

The mapping is supposed to be from $\mathbb{N}$ to $\mathbb{N}$. I'm still trying to understand if this is possible, I mean if it was from $\mathbb{R}$ to $\mathbb{N}$, I guess $x^2$ would work.
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1answer
29 views

Question on abstract algebra about invertiblity?

Let $(G,*)$ be a group. Could anyone give me an example $a,b\in G$ such that $$a*b=e ~ and ~ b*a\neq e $$ Where e is the identity element. I would appreciate any help. Thanks in advance!
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4answers
68 views

What would be an example of a magma such that $x\cdot(x\cdot x)\neq (x\cdot x)\cdot x$?

Let $(M,\cdot)$ be a non-associative magma. What would be an example of $M$ such that there exists $x\in M$ such that $x\cdot(x\cdot x)\neq (x\cdot x)\cdot x$?
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1answer
54 views

The strong operator limit of a sequence of unitary operators

If $\mathcal H$ is a Hilbert space and $U_n \in B(\mathcal H)$ is a strong-operator convergent sequence of unitary operators, say $U_n\rightarrow U$, is it true that $U$ is unitary? More explicitly, ...
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1answer
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Finding a counterexample: If $f=g^2$ and $f$ is differentiable on $[a,b]$, then $g$ is differentiable on $(a,b)$.

Suppose $f,g:[a,b]\to \mathbb R$. Provide a counterexample: If $f=g^2$ and $f$ is differentiable on $[a,b]$, then $g$ is differentiable on $(a,b)$. I've been attempting to find a counterexample by ...
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1answer
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1
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1answer
50 views

Counterexample of “the product of open subsets is open in a topological ring”?

Given a topological ring $R$ and $U,V$ open subsets, we can show that $U+V$ is an open subset due to the fact that $x\mapsto x+y$ is a homeomorphism for every $y \in R$. Since, in general, $R$ is not ...
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3answers
68 views

Class of algebras defined by quasi-identities is not closed under taking the quotients

Let $\mathbf{A}$ be an algebra (in the sense of universal algebra) of some signature $\Sigma$. By quasi-identity I mean the formula of the form $$(\forall x_1) (\forall x_2) \dots (\forall x_n) ...
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2answers
102 views

Introductory example(s) of a functor that is full but not faithful

What is your favourite example to offer real beginners of a functor which is full but not faithful?
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0answers
28 views

Is the conjugacy map between two distinct circle homeomorphisms unique?

Suppose $f,g,h_1,h_2$ are circle homeomorphisms with $f≠g$ and $fh_i = h_ig$ for $i=1,2$. Does it follow that $h_1 = h_2$? I restrict $f≠ g$ because I noticed that if $$f(x) := g(x) := R_\alpha(x) ...
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1answer
44 views

On convergence a.e and convergence measure

I have a question. First, I know that convergence in measure of a sequence of functions $f_n$ is different than convergence a.e., wich means there are sequences that converge in measure but not a.e. ...