0
votes
1answer
24 views

Various convergences in the space of bounded operators

Could you please help me to find some classical (counter)examples in functional analysis? Let $X$ and $Y$ be some normed spaces over $\mathbb{C}$. By $\mathcal{B}(X,Y)$ we denote the space of bounded ...
6
votes
2answers
71 views

A function such that $f'(x)>0$, but not strictly monotone increasing.

This is Exercise 10.3.5 from Analysis Vol.1 by Terence Tao. Give an example of a subset $X \subset \mathbb{R}$ and a function $f: X \to \mathbb{R}$ which is differentiable on $X$, is such that ...
-1
votes
1answer
23 views

Some Topological Properties of Starlike Sets!

A subset $E$ of $\mathbb R^n$ is starlike if it contains a point $p_0$ (called a center for $E$) such that for each $q\in E$, the segment between $p_0$ and $q$ lies in $E$. For more information please ...
2
votes
2answers
55 views

Example of Localization and Prime Ideals

For each $n\in \mathbb Z^+$, give an example of a localization of $\mathbb Z$ with exactly $n$ prime ideals. Justify your answer. Could an example have something to do with a UFD or Noetherian ...
0
votes
3answers
49 views

Discrete HW Question

Show that For all sets A and B, $A^{C} \cup B^{C} \subset (A \cup B)^{C}$ is false by a counterexample.
1
vote
1answer
50 views

Counterexample of separation theorem

I'm trying to know a counterexample for separation theorem: If $A$ and $B$ are two disjoint convex set in a topological vector space $X$, one of them has nonempty interior, then there exists $f\in ...
1
vote
0answers
35 views

nearest point and closed complement of a subspace in norm spaces

It is well-known that in any Hilbert space $H$, each closed subspace $Y$ admits a closed complement $Y^\perp$. This result also implies that there exists a best approximation point to $Y$ for any ...
3
votes
1answer
146 views

A calculus counterexample!

Give me an example of two Riemann-integrable functions $f,g:[0,1]\to[0,1]$ such that $g\circ f$ isn't integrable! I already know the following example: $$f(x)=\begin{cases} 0, & \text{if ...
5
votes
2answers
68 views

Two non-homeomorphic connected, hausdorff, locally compact spaces whose one-point compactifications are homeomorphic

I'm looking for two non-homeomorphic connected, Hausdorff, locally compact spaces whose one-point compactifications are homeomorphic. Without the connectedness property this is easy, for example: ...
0
votes
1answer
28 views

Simple question about montonically increasing function

Suppose we have a continuous $f: \mathbb{R} \to \mathbb{R}$, we know the definition of a monotonically increasing function is for $x,y \in \mathbb{R}$, if $x < y$ then $f(x) < f(y)$. I know that ...
4
votes
1answer
48 views

Topological space in which there are no close and compacts subsets (except for the empty set)

Any example of those topological spaces? I cant think of no one :S I think it must be infinite and it must not be T2, but no idea how to find one.
0
votes
1answer
96 views

Isometry of a metric space with proper subset

In Irving Kaplansky's "Set Theory and Metric Spaces", exercise 17 on page 71 asks for an example of a metric space which is isometric to a proper subset of itself. Any infinite discrete space and any ...
0
votes
2answers
82 views

Controllability properties of discrete vs. continuous systems

I'm not sure what differentiates discrete from continuous systems in trying to prove certain properties of controllability. I have a chain of proofs from class that prove each other for a continuous ...
2
votes
1answer
50 views

Basic Multilinear regression question for finding examples or counterexamples.

Hello Wise mathematicians! I have few quenstions about Multi linear regresstion. I've been asked from my friend, but I have very weak knowledge background from that field. It seems my friend is in ...
0
votes
2answers
227 views

locally compact Hausdorff space which is not second-countable

I'm trying to find an example of a space that is Hausdorff and locally compact that is not second countable, but I'm stuck. I search an example on the book Counterexamples in Topology, but I can't ...
1
vote
1answer
2k views

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 ...
0
votes
5answers
79 views

Operator with symmetric but without associative?

Addition and multiplication in math, all is symmetric,associative。 But i have no idea about operators with symmetric but without associative. please help me listing any 2-arity operators?
4
votes
2answers
167 views

Looking for a counter example for non-connected intersection of descending chain of closed connected sets

Let $X$ be a topological space and let $\left\{ Y_{i}\right\} _{i=1}^{\infty}$ be a descending chain of closed connected subsets of $X$. I know from reading elsewhere that ${\displaystyle ...
1
vote
1answer
52 views

Show that floor function does't satisfy FTC.

The function is $f(x) = \lfloor 1-x^2 \rfloor$. $$f(x) = \left \{ \begin{array}{lr} -3 & : x \in [-2,-\sqrt{3})\\ -2 & : x \in [-\sqrt{3},-\sqrt{2})\\ -1 & : x ...
9
votes
2answers
914 views

Give an example of a function $h$ that is discontinuous at every point of $[0,1]$, but with $|h|$ continuous on $[0,1]$

Give an example of a function $h:[0,1]\to\mathbb{R}$ that is discontinuous at every point of $[0,1]$, but such that the function $| h |$ that is continuous on $[0,1]$. I don't really even know where ...
1
vote
1answer
54 views

A question on isolated point

Is a star countable space $X$ with at most countable non-isolated points always Lindelof? A topological space $X$ is said to be star countable if whenever $\mathscr{U}$ is an open cover of $X$, ...
1
vote
0answers
52 views

Non context-free languages closed under reverse

Is this sentence true or false? I'll be glad for some explaination L is not context-free, then its reverse is also not context-free Thanks in advance
1
vote
1answer
52 views

Example ideal of $\mathfrak{sl}(2,\mathbb{C})$

I need an example about ideal from lie algebra $\mathfrak{sl}(2,\mathbb{C})$ except trivial ideal and $\mathfrak{sl}(2,\mathbb{C})$ itself, can someone help me? I try to make ideal except trivial ...
1
vote
2answers
179 views

An epimorphism in $\text{Grp}$ without right inverse?

Exercise 8.24 in Aluffi's Algebra: Chapter 0 asks us to find an epimorphism in $\text{Grp}$ without right inverses. I happen to know that epimorphisms in $\text{Grp}$ are surjective, so we need a ...
3
votes
1answer
87 views

A countale partially ordered set that has an uncountable number of maximal chains

I'm looking for a countable set S with a partial order < that has an uncoubtable number of maximal chains. I had many ideas but non of then is correct (for example- S= natural numbers, "<" is ...
1
vote
4answers
1k views

iid variables, do they need to have the same mean and variance?

If two random variables $x$ and $y$ are identical and independently distributed, do they need to have the same mean and variance? Can there exist a case where they are iid and still have different ...
3
votes
1answer
222 views

Examples of preorders in which meets and joins do not exist

Exercise 1.2.8 (Part 1), p.8, from Categories for Types by Roy L. Crole Definition: Let $X$ be a preordered set and $A \subseteq X$. A join of $A$, if such exists, is a least element in the set of ...
4
votes
1answer
127 views

On $T_2$, first countable, countably compact space

As we know, For every $T_2$, first countable, compact space, its cardinality is not more than $2^\omega$. (See chapter 3 of Engelking's book.) However, I want to know whether the result is ...
3
votes
2answers
116 views

Sequence convergence and parentheses insertion

find an example for a series $a_{n}$ that satisfies the following: $a_{n}\xrightarrow[n\to\infty]{}0$ ${\displaystyle \sum_{n=1}^{\infty}a_{n}}$ does not converges There is a way to insert ...
2
votes
0answers
119 views

Is inversion sequentially continuous in SOT?

Let $A_n \overset{SOT}{\to} A$ where $A$ is invertible. Does $A_n^{-1} \overset{SOT}{\to} A^{-1}$? Does $A_n^{-1} \overset{WOT}{\to} A^{-1}$? EDIT: Forgot to mention $\{A,A_n\}\in\mathscr{B(H)}$ ...
5
votes
1answer
232 views

Is my counter-example correct?

In my homework for real-analysis I was asked to prove the following statement: On $[0,1]$, for $1\leq{}p<\infty$, If $f_{n}\rightarrow{}f$ a.e. and $||f_{n}||_{p}\leq{}M \space\space\forall\space ...
3
votes
2answers
887 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 ...
2
votes
2answers
84 views

A relation $R$ such that $R\cup R^{-1}$ is not an equivalence relation

I have a homework assignment to find a Relation $R$ over $A = \{1,2,3\}$ where $R\cup {{R}^{-1}}$ is not an equivalence relation (transitive, reflexive and symmetrical). $R$ must be Transitive and ...
4
votes
1answer
517 views

Continuous bijections from the open unit disc to itself - existence of fixed points

I'm wondering about the following: Let $f:D \mapsto D$ be a continuous real-valued bijection from the open unit disc $\{(x,y): x^2 + y^2 <1\}$ to itself. Does f necessarily have a fixed point? I ...