Mathematical analysis. Consider a more specific tag instead: (real-analysis), (complex-analysis), (functional-analysis), (fourier-analysis), (measure-theory), (calculus-of-variations), etc. For data analysis, use (data-analysis).

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1answer
94 views

A taylor series for an integral with a singularity

Suppose $f: \mathbb{R} \rightarrow \mathbb{R}$ is a smooth function with a single root we call $y_*$. Then define \begin{equation} F_{\delta}:=\int^{y_*+ \delta/2}_{y_*- \delta/2} 1/f(y)dy ...
4
votes
3answers
187 views

Prove $f(x) = 0 $for all $x \in [0, \infty)$ when $|f'(x)| \leq |f(x)|$

mathematicians! I want to ask to all wise people about a problem I met at the quiz to obtain some ideas. The problem is following. It may not be accurate since the problem is dependent on my memory ...
0
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1answer
46 views

Question about a problem solution.

Let us define $$ \int\limits_E f \, dm = \sup Y( E, f ) $$ where $Y(E,f) = \left\{ \int\limits_E \varphi \, dm: 0 \leq \varphi \leq f \right\}$ where $\varphi$ is a simple function. Here is the ...
3
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2answers
42 views

How to check whether a given inequality is correct for a large span of integers?

The inequality $\sqrt{n+ 1}−\sqrt n < \frac{1}{\sqrt n}$ is false for all n such that $101 ≤ n ≤ 2000$. Is the statement true?
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2answers
54 views

Form of a harmonic function

Given that $\phi(x^2+y^2)$ is harmonic, where $\phi: (0, \infty)\to \mathbb{R}$, find the form of $\phi$. I do not know what they mean by form nor could I find anything online... My book says that ...
2
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1answer
75 views

Showing that the preimage of a continuous function on R is a σ-algebra

Let $f$ be a continuous function on $\mathbb{R}$. Define $\mathcal{A}=\left \{ E\subseteq \mathbb{R} : f^{-1}(E)\in \mathcal{B}(\mathbb{R})\right \}$. I want to show that $\mathcal{A}$ is a ...
2
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1answer
327 views

Clarification on this corollary of the Arzela-Ascoli Theorem

I am given the following corollary without proof: A family of continuous functions on a compact metric space into $\mathbb R^m$ is compact iff it is closed, equicontinuous and bounded. Does ...
2
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1answer
146 views

Trying to Understand Baby Rudin Proof

If p is a limit point of a set E, then every neighborhood of p contains infinitely many points of E Proof: Suppose there is a neighborhood N of p which contains only a finite number of points of E. ...
2
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3answers
137 views

Different limits for the alternating harmonic series?

Show that the series $$\sum_{n=1}^{\infty} \dfrac{(-1)^n}{n}$$ is not absolutely convergent. Show that by permuting the terms of the series one can obtain series with different limits. I am able to ...
0
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1answer
33 views

Question on the convergence of $|x|^n\rightarrow 0$ when $|x|<1$.

My question is about calculating the $N$ for a convergent sequence. I want to prove that $|x|^n \rightarrow 0$ when $|x|<1$ using the $\epsilon, N$ definition. So I know that $|x^n-0| \leq ...
1
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1answer
135 views

Question on a corollary of the Arzela-Ascoli theorem

I am given a corollary of the Arzela-Ascoli theorem, and I've substantially rephrased it to this: If $S$ is an equicontinuous and pointwise bounded set of functions with domain a compact metric ...
2
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2answers
101 views

What does it mean for a set to be compact in another set?

I am given the following definition: Let $B$ be a set of continuous maps with domain a metric space $A$ and codomain a metric space $N$, and $B_x=\{f(x):f\in B\}$. $B$ is pointwise compact ...
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2answers
160 views

$f(X \cap Y) \subset f(X) \cap f(Y) $

In class we had the following function which I intend to prove for my own peace of mind. Let $M$ and $N$ be sets and $f: M \longrightarrow N$ a function: \begin{align}f: P(M) &\longrightarrow ...
1
vote
1answer
59 views

Is $\{ X(A) : A \in \mathcal{F} \} $ a $\sigma$ - Algebra?

Let $( \Omega, \mathcal{F}, P )$ be a probability space a let $X : \Omega \to \mathbb{R} $ be a random variable. My question is: Is $\{ X(A) : A \in \mathcal{F} \} $ a $\sigma$ - Algebra ? My ...
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4answers
151 views

How to prove that $a<S_n-[S_n]<b$ infinitely often

Let $S_n=1+\frac{1}{2}+\frac{1}{3}+\ldots+\frac{1}{n}$, where $n$ is a positive integer. Prove that for any real numbers $a,b,0\le a\le b\le 1$, there exist infinite many $n\in\mathbb{N}$ such that ...
4
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2answers
135 views

If $f^{-1}(G)$ is open in $X$ for every open set $G$ in $Y$, then $f$ is continuous. Question on proof.

Let $X,Y$ be metric spaces and $f:X\rightarrow Y$. If $f^{-1}(G)$ is open in $X$ for every open set $G$ in $Y$, then $f$ is continuous. The text I am using proves this proposition like so: Suppose ...
3
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1answer
392 views

Show convergence of a sequence of continuous functions $f_n$ to a continuous function $f$ does not imply convergence of corresponding integrals.

Let $f_n\in C([0,1])$ be a sequence of functions converging uniformly to a function $f$. Show that $$\lim_{n\rightarrow\infty}\int_0^1f_n(x)dx = \int_0^1 f(x)dx.$$ Give a counterexample to show that ...
2
votes
1answer
192 views

Sufficient and necessary conditions to get an infinite fiber $g^{-1}(w)$

I want to verify the proof of this result and get some start ideas to overcome the different steps of this proof. Lemma: Let $g$ be a real analytic function. Then we have the equivalence ...
3
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1answer
39 views

Surjectivity of a function considering $f \circ h=Id_Y$

\begin{align} f: \mathbb{R}^2 &\longrightarrow \mathbb{R} \\ (x,y) & \longmapsto x+y \end{align} Question: Is this function surjective? It seems clear to me that this function must be ...
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1answer
121 views

Abstracting Completeness

Introduction This may seem like a weird question or even a silly one, but topology is vast and I find I make quicker progress working my way through it by trying ideas out loud within earshot of ...
1
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1answer
624 views

Questions about coercive functions and its implications

Given this definition: A function $f:\mathbb{R}^n\rightarrow \mathbb{R}$ is $coercive$ if $$\lim_{||x||\rightarrow\infty}f(x) = \infty.$$ Explicitly, this means that for any $M>0$ there is an ...
1
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1answer
77 views

regularity of the matrix exponential

For every matrix $A\in \mathbb{R}^{m^2}$ let $$e^{A}=\sum\limits_{k=0}^{\infty }\frac{A^{k}}{k!}$$ prove that the application $exp: A \rightarrow e^A$ is $C^{\infty}$ class. Thanks beforehand.
3
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1answer
86 views

$F = \{f\in C^1([0,1])| \hspace{2mm} \|f\|\leq M, \|f'\|\leq N\}$. Showing it is precompact and not closed.

I have an example in my book: Let $C([0,1])$ denote the space of all continuous functions $f$ on $[0.1]$ with continuous derivative $f'$. For constants $M>0$ and $N>0$, we define the subset $F$ ...
3
votes
1answer
66 views

Is this inequality true? (Inequality involving probability distribution and products)

Suppose $f(z)$ is a discrete probability distribution with space $S$. Suppose $g(z),h(z)>0$ for all $z \in S$. Is it true that $$\prod_{z \in S}{g(z)^{f(z)}}+\prod_{z \in S}{h(z)^{f(z)}} \leq ...
4
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1answer
2k views

Prove that $C^1([a,b])$ with the $C^1$- norm is a Banach Space

Consider the space of continuously differentiable functions, $$C^1([a,b]) = \{f:[a,b]\rightarrow \mathbb{R}|f,f' \text{are continuous}\}$$ with the $C^1$-norm $$||f|| = \sup_{a\leq x\leq ...
1
vote
1answer
99 views

The projection on the first factor is a bijection

I require some clarification and hints on the following Problem: \begin{align}f: X \longrightarrow Y \end{align} The image $p$ is defined as: \begin{align} p: G_f &\longrightarrow X \\ (x,y) ...
1
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1answer
289 views

On compositions $g \circ f$ and whether $f$ is surjective

\begin{align} f: X \longrightarrow Y \\ g: Y \longrightarrow Z \\ g \circ f: X \longrightarrow Z \tag{is bijective} \end{align} The bijective conditions only applies to $ g \circ f$. I already managed ...
1
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1answer
99 views

Palais-Smale and Cerami

I have this two definitions : 1) "$\phi$ satisfies the Palais-Smale compactness condition at the level c, or $(PS)_c$ for short, if every sequence $(u_j)\subset W$ such that $\phi(u_j)\rightarrow c$, ...
0
votes
2answers
50 views

On compositions of functions and their properties

Given: $ f: X \longrightarrow Y\\ g: Y \longrightarrow Z $ and $g \circ f : X \longrightarrow Z $ is a bijection. I want to analyse wether $f$ is an injection, a surjection or both (a bijection). ...
3
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1answer
78 views

A query about countability

Suppose we are working in an elementary context where we want to keep background assumptions modest (not take a stand on fancy issues in set theory, say). What should our attitude be to the idea of ...
4
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0answers
120 views

On the sequence $(f_n)$ defined by $f_1(x)=x$ and $f_{n+1}(x)=x^{f_n(x)}$

Consider the numbers $x^x$,$x^{(x^x)}$,$x^{(x^{(x^x)})}$, etc. Let $n$ be the number of times $x$ appears in the power tower and $f_n$ the corresponding function, for example ...
3
votes
1answer
276 views

Let $X$ be a compact metric space. If $f:X\rightarrow \mathbb{R}$ is lower semi-continuous, then $f$ is bounded from below and attains its infimum.

Let $X$ be a compact metric space. If $f:X\rightarrow \mathbb{R}$ is lower semi-continuous, then $f$ is bounded from below and attains its infimum. I want to prove this. This is my proof: Since $X$ ...
1
vote
1answer
132 views

Describe explicitly the $M$-measurable functions in case $M$ is one of the following $\sigma$-algebras:

Describe explicitly the $M$-measurable functions in case $M$ is one of the following $\sigma$-algebras: (a) $M=\{\emptyset,X\}$ (b) $M=2^{X}$ (c) For certain disjoint sets $E_1,...,E_N$, ...
0
votes
0answers
95 views

Prove that a set $A\subset\mathbb{R}^n$ is measurable $\iff$ there exist a set $B$ which is an $F_{\sigma}$ and a set $C$ which is a $G_{\delta}$.

Prove that a set $A\subset\mathbb{R}^n$ is measurable $\iff$ there exist a set $B$ which is an $F_{\sigma}$ and a set $C$ which is a $G_{\delta}$ such that $B\subset A\subset C$ and $C$~$B$ (C without ...
4
votes
1answer
311 views

Prove that if $N$ is a null set in $\mathbb{R}^n$, then there exists a Borel null set $N'$ such that $N\subset N'$.

Prove that if $N$ is a null set in $\mathbb{R}^n$, then there exists a Borel null set $N'$ such that $N\subset N'$. In fact, prove that $N'$ may be chosen to be a $G_{\delta}$, a countable ...
5
votes
1answer
100 views

$P(X)$ is locally compact if $X$ is?

Let us assume, as in here Measurable structure on the space of probability measures that $X$ is a locally compact Polish space. Then can the same thing be said of $P(X)$, its probability measures ...
0
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1answer
32 views

a problem about Inequality zoom

I came across this inequality. Can anyone help me?
2
votes
2answers
142 views

Determine the set of points where $f$ is continuous.

Define $f:[0,1]\rightarrow\mathbb{R}$ by $$ f(x) = \begin{cases} x & \text{if $x$ is irrational} \\ p\sin(\frac{1}{q}) & \text{if x=$\frac{p}{q}$, where $p,q$ are relatively prime integers.} ...
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1answer
309 views

The mapping of a Set onto the characteristic function is a bijection [duplicate]

Let $X$ be a Set. For all Subsets $ A \subset X$ the characteristic function of A is defined as: \begin{align} \chi_A(x)= \begin{cases} 1 \iff x \in A \\ 0 \iff x \notin A \end{cases} \end{align} Let ...
0
votes
1answer
44 views

Comparison of Metric Spaces and Completeness

Yesterday I had a discussion in which I presented a proof that a particular metric space is not complete. On this space $X$ there are two metrics $d$ and $d'$, with respect to $d'$ it is not complete, ...
1
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2answers
74 views

How to compare two functions which satisfy some conditions

$f(x),g(x)$ are defined on $[-1,1]$, $f'(0),g'(0)$ exist, $f(0)=g(0)$, and $f(x)\ge g(x)$ holds for an open interval containing $0$. Then which of the following is correct: I, $f(x)$ and $g(x)$ have ...
8
votes
1answer
431 views

How to prove that there exists $g(x)$ such $\int_{0}^{1}g(x)dx\ge\frac{1}{2}\int_{0}^{1}f(x)dx$

let $f(x)\ge 0,x\in [0,1]$, and is increasing in $[0,1]$ show that: There exists $g(x)\ge 0,x\in [0,1]$,and $g''(x)>0$, such $g(x)\le f(x)$, and such ...
1
vote
1answer
52 views

Periodicity of a function

If f(x) is periodic with period a, would f(tx) be periodic with period a/t? Would f(tx+b) make still have period a/t? Im inclined to think so, because this works for the trig functions, but i'm not ...
2
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1answer
94 views

Parametrization of level sets of a smooth function

Let $H:\mathbb{R}^2\rightarrow\mathbb{R}$ be given by $H(q,p)=p^2/2+3q^2/2$ (single-well potential). This function has a critical point at $(0,0)$. Define $T:\mathbb{R}^+\rightarrow \mathbb{R}$ by, ...
4
votes
1answer
140 views

Measurable structure on the space of probability measures

My advisor only half-jokingly mentioned that sometimes people like to consider the measurable structure on $P(X)$ where X is a locally compact polish space and $P(.)$ denotes the probability measures ...
2
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0answers
41 views

Surjective local diffeomorphism not injective

I would like to find a function $F:\mathbb{R}^N\to\mathbb{R}^N$, for $N\geq 2$, which is surjective and a local diffeomorphism, but it is not injective. I can solve the problem by using partition of ...
0
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1answer
65 views

Any material available on generalized power series?

I'm in my final year in my undergrad, and part of my thesis will involve generalized power series. Does anyone know of any good resources out there for learning about them? I struggled to find ...
0
votes
1answer
100 views

Justification of termwise integration

Please explain how the final line of the following proof implies the result of the corollary? (Note that the main middle section of the proof is for a preceding theorem.)
3
votes
3answers
3k views

Commutator of $x$ and $p^2$

I have a question: If I have to find the commutator $[x, p^2]$ (with $p= {h\over i}{d \over dx} $) the right answer is: $[x,p^2]=x p^2 - p^2x = x p^2 -pxp + pxp - p^2x = [x,p]p + p[x,p] = 2hip$ But ...
1
vote
2answers
66 views

Bijections and disjoint sets by finite summation

Suppose $X$ is finite and $f:X\rightarrow\mathbb{R}$ a function. Let $g:Y\rightarrow X$ be a bijection. Then $$\sum_{x\in X} f(x) =\sum_{y\in Y} f(g(y))$$ That is what I first want to prove. The ...