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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5
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2answers
137 views

Problem connecting Topology and Algebra via Analysis

Let $C(X):=$ Set of all complex/real valued continuous functions. If $X$ is compact then all the maximal ideals in the ring $C(X)$ is of the form $M_{x}=\{f\in C(X): f(x)=0\}$ for some $x\in X$. Is ...
0
votes
2answers
95 views

A metric space problem [duplicate]

Possible Duplicate: Example to show the distance between two closed sets can be 0 even if the two sets are disjoint Let $(X,d)$ be a metric space and $A,B$ be two distinct closed set in $X$ ...
1
vote
1answer
142 views

power series estimate (convergence)

Let $f(x)=\sum\limits_{n=0}^\infty a_nx^n$ a power series and $f(0)\ne0$. (w.l.o.g. $f(0)=1$) Suppose the power series has radius of convergence $r>0$. A power series is continuous in her ...
0
votes
2answers
190 views

Why Jordan measure is undefined?

$R^2$, $A=\{(x,\;y)\in R^2\colon 0\leqslant x\leqslant 1,\;0\leqslant y\leqslant 1\}$. Consider $X=A\cap Q^2$. Why for $X$, $m_e X=1,\;m_i X=0,\;m_e X\neq m_i X$? Especially i interested in why inner ...
1
vote
0answers
261 views

Exercise: continuity of a function of two variables

Consider a continuous function $\phi: \mathbb{R}^n \times \mathbb{R}^m \rightarrow \mathbb{R}_{\geq 0}$ and a locally bounded function $\psi:\mathbb{R}^n \rightarrow \mathbb{R}^m$. So we study ...
1
vote
2answers
231 views

Cauchy sequence alternative definition.

Is it possible to define a Cauchy sequence as follows? Let $(X,d)$ be a metric space and $(x_{n})_{n\in \mathbb{N}}$ be a sequence in it. Then $(x_{n})_{n\in \mathbb{N}}$ is Cauchy iff ...
1
vote
1answer
142 views

convergence of power series

We have given a power series $\sum\limits_{n=0}^\infty a_nx^n$ and $a_0\ne0$. Suppose the power series has radius of convergence $R>0$. Then we may assume that $a_0=1.$ Explanation: It's ...
4
votes
3answers
908 views

If $f(x + y) = f(x) + f(y)$ showing that $f(cx) = cf(x)$ holds for rational $c$

For $f:\mathbb{R}^n \to \mathbb{R}^m$, if $f(x + y) = f(x) + f(y)$ for then for rational $c$, how would you show that $f(cx) = cf(x)$ holds? I tried that for $c = \frac{a}{b}$, $a,b \in \mathbb{Z}$ ...
0
votes
1answer
64 views

Non-finite series implies product is zero

Given $0 \le y_n \le 1$ and $\sum_{n \in \mathbb{N}} y_n = \infty$, how can we show $\prod_{n=1}^\infty (1 - y_n) = 0$?
3
votes
3answers
122 views

To prove $f(x)\to\infty$ with an “Oresme” strategy

My goal is to prove that: $\displaystyle\sum\limits_{n=1}^\infty \frac{1}{n^{x}} \rightarrow \infty$ when $ x\rightarrow 1^+$ My first approach (which failed) is here: To prove ...
3
votes
1answer
114 views

To prove $f(x)\rightarrow \infty$ with a “home made” strategy

My goal is to prove that: $\displaystyle\sum\limits_{n=1}^\infty \frac{1}{n^{x}} \rightarrow \infty$ for all $ x\rightarrow 1^+$ In order to show this statement I show that no matter how big ...
0
votes
2answers
53 views

Convex function with linear grow?

I'm looking for a continuous, strictly increasing, strictly convex function $f: \mathbb{R}_{\geq 0} \rightarrow \mathbb{R}_{\geq 0}$, with $f(0)=0$, and such that $$ \lim_{x \rightarrow\infty} ...
2
votes
0answers
60 views

$f(x) \rightarrow \infty $ when $x\rightarrow 1^+$

I want to prove that $f(x) \rightarrow \infty $ when $x\rightarrow 1^+$. My tactic is to prove that no matter how big you choose a $N\in \mathbb{R}$, you can always find a $\delta>0$ so the ...
0
votes
2answers
86 views

Two Analysis Questions

1) Define : $\langle z\rangle := (1+|z|^2 ) ^\frac{1}{2} $ for all $z \in \mathbb{C} $. Prove : $\langle x+y\rangle \leq 2\langle x\rangle\langle y\rangle $ for all $x,y \in \mathbb{R} ^N$ . 2) ...
3
votes
1answer
89 views

inequality in a differential equation

Let $u:\mathbb{R}\to\mathbb{R}^3$ where $u(t)=(u_1(t),u_2(t), u_3(t))$ be a function that satisfies $$\frac{d}{dt}|u(t)|^2+|u|^2\le 1,\tag{1}$$where $|\cdot|$ is the Euclidean norm. According to ...
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vote
2answers
1k views

$\mathcal{C}^1$ implies locally Lipschitz in $\mathbb{R}^n$

According to wikipedia a function $f\colon \mathbb{R}^n\to\mathbb{R}^n$ that is continuously, is also locally Lipschitz. I there someone who knows a good reference which contains a proof of this ...
2
votes
0answers
225 views

calculating the amplitude of a cosine function

I want to be able to be able to get the amplitude of the following function: $$||A||\cos(2 \omega t + a)+||B||\cos(3 \omega t +b)+||C||\cos(5 \omega t +c)$$ I am trying to find a way to get the ...
2
votes
1answer
868 views

properties of a real analytic function

If there are a radius $r>0$ and constants $M,C\in\mathbb R$ for all $y\in U$ with $$|\partial^if(x)|\leq M\cdot i!\cdot C^{|i|}\space\space\space\space \forall x\in\mathbb B_r(y),i\in\mathbb ...
3
votes
1answer
108 views

Approximating the logarithm of a Laplace transform

Suppose $X$ is a random variable on $\mathbb R_+$ with finite mean, i.e. $\mathbb E X <+\infty$. Let $F_X(t)$ be its c.d.f. and $\mathcal{L}_X(\cdot)$ its Laplace transform, i.e. ...
10
votes
1answer
578 views

Riemann's thinking on symmetrizing the zeta functional equation

In the translated version of Riemann's classic On the Number of Prime Numbers less than a Given Quantity, he quickly derives the zeta functional equation through contour integration essentially as ...
2
votes
0answers
43 views

Optimizing a Composed Function

Let $f: \mathbb{R}^n \rightarrow \mathbb{R}_{\geq 0}$ be continuous. Consider $g: \mathbb{R}_{\geq 0} \rightarrow \mathbb{R}_{\geq 0}$ strictly increasing, continuous and such that $g(0)=0$. I think ...
2
votes
1answer
72 views

Exercise: optimization and limit

Let $\underline{a}_N:=(a_1,...,a_N)^\top \in \mathbb{R}^N$ denote a vector of lenght $N$. Define $$ x_N^* := \arg \min_{x \in \mathbb{R}} \left\| \underline{a}_N - x \mathbb{1}_N \right\|_p $$ where ...
6
votes
1answer
118 views

Showing $f\left(A\right)=e^{A^{2}}$ is differentiable.

Let $f\left(A\right)=e^{A^{2}}$ where $A$ is an $n\times n$ matrix. Show that $f$ is differentiable and compute its derivative. I know this is kind of a basic question, but I am not sure how ...
2
votes
1answer
148 views

Smoothness of Fourier series

In a book from differential equations I found the following theorem, without proof and references: Let functions $f, g: R \rightarrow R$ be continuous and $2\pi$-periodic and let $m\in N$. Assume ...
2
votes
1answer
116 views

Continuity of Parametric Integral

Consider a continuous function $f: X \times Y \rightarrow \mathbb{R}_{} \geq 0$, where $X \subset \mathbb{R}^n$ is compact, and $Y \subseteq \mathbb{R}^m$ is closed. Define $\hat{f}:X \rightarrow ...
0
votes
1answer
319 views

Redefining Outer Lesbegue Measure on $\Bbb{R}^{d}$ From Closed Cubes to Rectangles.

UPDATE: I added an answer based off the hints provided by copper.hat. It may, however, need some adjustment. I'm trying to solve another question from Stein and Shakarchi's analysis text. ...
1
vote
1answer
139 views

Continuity of Expected Value

Let $m(\cdot)$ be a probability measure on $Z$, so that $\int_Z m(dz) = 1$. Consider a continuous function $f: X \times Y \times Z \rightarrow \mathbb{R}_{\geq 0}$, where $X \subseteq \mathbb{R}^n$, ...
2
votes
1answer
59 views

Series basic question

$$\sum^N_{n=1}\liminf_{k \to \infty} f_k(n) = \lim_{k \to \infty} \sum_{n=1}^N \inf_{j \ge k} f_j(n)$$ I am not sure that equation true. Is that equation true? Then why is it?
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vote
0answers
348 views

Tensor product of the space of smooth functions

The Hilbert Space tensor product gives $L^2(\mathbb R^2,dx\otimes dx;\mathbb R)= L^2(\mathbb R,d x;\mathbb R) \otimes L^2(\mathbb R,dx;\mathbb R)$ My question is: does there exist also a notion of ...
4
votes
1answer
218 views

About the remainder of Taylor expansion and Riemann-Liouville integral

Integral form of Taylor expansion looks like this:$$f(x)=\sum_{i=0}^k\frac{f^{(i)}(a)}{i!}(x-a)^i+\int_a^x\frac{f^{(k+1)}(t)}{k!}(x-t)^kdt$$ Riemann-Liouville integral is ...
2
votes
2answers
132 views

Rolle's Theorem on a derivative…

I have the following: Suppose $p(x) =(x^{2}-1)^{m}$ for some $m \geq2$. Applying Rolle's Theorem on each of the intervals $[-1,c]$ and $[c,1]$, show that $p''(x)$ has at least two roots in ...
1
vote
1answer
93 views

Characterization of strong minimums with slices.

I am doing a proof of a Lemma that isn't in a book. Let $X$ a Banach space and $\emptyset\not=S\subset X$ closed of $X$. Let $f$ be a lower semicontinuous function bounded below in $S$. I have that ...
2
votes
2answers
185 views

Want f differentiable at the origin but discontinuous everywhere else!

I am trying to get a function $f:\mathbb{R}^2 \to \mathbb{R}$ that is differentiable at the origin but discontinuous everywhere else? As a simpler case, we have that $$g\left(x\right)=\begin{cases} ...
3
votes
1answer
602 views

Showing that the characteristic function of $\Bbb Q$ is not Riemann Integrable

So I know that the characteristic function of the rationals is not Riemann integrable and we can show this by showing that the upper and lower sums are different. But I have a theorem in my notes ...
2
votes
4answers
106 views

Limit to infinity question [duplicate]

Possible Duplicate: Finding the limit of $\frac{Q(n)}{P(n)}$ where $Q,P$ are polynomials Does $\lim_{x\rightarrow \infty} \frac{5x}{(1+x^2)} = 0$ or $\lim_{x\rightarrow \infty} ...
17
votes
3answers
917 views

Choice of $q$ in Baby Rudin's Example 1.1

First, my apologies if this has already been asked/answered. I wasn't able to find this question via search. My question comes from Rudin's "Princicples of Mathematical Analysis," or "Baby Rudin," ...
2
votes
3answers
679 views

WolframAlpha blows simple substitution?

I am to find a formula for the n-th derivative of $$\frac{1+x}{1-x}$$ I came up with $$\frac{(-1)^{n+1} \times 2 \times n!}{(1-x)^{n+1}}$$ This seems right, but I noticed that WolframAlpha somehow ...
1
vote
1answer
521 views

Big O Notation question

I am trying to understand the Big-O and little-O notation, so I plotted 2 graphs which I have posted below, but I still dont really get the concept of it. What exactly does the ...
1
vote
1answer
510 views

Hilbert Spaces and Closed Subspaces

Let $H$ be a Hilbert Space, and $M$ a closed subspace. Is it true that $H = M \bigoplus M^{\perp}$ Does this hold if $M$ is not closed? Or only if $H$ is finite/infinite dimensional?
7
votes
3answers
799 views

Examples of perfect sets.

Let $0\lt a\lt 1$. Can I get examples of of subsets of $[0,1]$ that are perfect sets, contains no intervals and has measure $1-a$. Well, I know by construction the Cantor set is perfect, contains ...
3
votes
1answer
105 views

Doubt about function value (expected undefined, but Wolframalpha says otherwise)

I have the function $f(x) = 1 + \frac{12x+4}{\left( x+1 \right)^2} \cdot \left( \frac{12}{12x+4} - \frac{2}{x+1} \right)$ (actually the derivate of another function, but that shouldn't matter). Since ...
0
votes
1answer
185 views

Bounding Expected Value

Let $\mu(\cdot)$ be a probability measure on the closed set $\mathbb{W} \subseteq \mathbb{R}^m$. Consider $f: \mathbb{X} \times \mathbb{W} \rightarrow \mathbb{R}_{> 0}$ locally bounded, where ...
4
votes
1answer
427 views

Difference between soft analysis and hard analysis

I have sometimes overheard people using the terms hard analysis and soft analysis.I am not a particularly well-read person in mathematics but I have wondered what that is all about.I hope there ...
2
votes
0answers
51 views

About function which Fourier coefficients satisfy $a_n=o(n^{-2}), b_n=o(n^{-2})$

Assume that a function $f: R\rightarrow R$ is $2 \pi$ -periodic and integrable on $[ -\pi,\pi] $. Let $(a_n)$, $(b_n)$ are its Fourier coefficients and $n^2 a_n, n^2 b_n \rightarrow 0$. Then by ...
1
vote
1answer
1k views

Showing that $\cos(x)$ is a contraction mapping on $[0,\pi]$

How do I show that $\cos(x)$ is a contraction mapping on $[0,\pi]$? I would normally use the mean value theorem and find $\max|-\sin(x)|$ on $(0,\pi)$ but I dont think this will work here. So I think ...
4
votes
1answer
101 views

$K$ compact and $\Omega$ is open, then $\inf\{\rho(x,x') \mid x \in K \textrm{ and } x' \in \Omega^c\} > 0$

I have to show the following: $(V,\rho)$ be a metric space, $K\subset V$ compact and $\Omega \subset V$ is open, then $d(K,\Omega^c) = \inf\{\rho(x,x') \mid x \in K \textrm{ and } x' \in \Omega^c\} ...
1
vote
0answers
190 views

A metric space is path connected and countable then it is complete

I have to show that if a metric space is path connected and countable then it is complete. I'm pretty lost where to start this at all. I have the basic definitions of complete, path-connected, compact ...
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vote
0answers
225 views

Strongly exposed points/Exposed points

I was studying and I got the next doubt: We suppose that $(X,\|\cdot\|)$ is a Banach space and $C$ it is a convex closed subset of X. We say that $x\in C$ it is an exposed point of $C$ if $\exists ...
5
votes
1answer
339 views

An Exercise on Inverse Function Theorem

Consider the mapping $f:R^2\rightarrow R^2$ given componentwise by: $f_1(x,y)=x+a_1x^2+2b_1xy+c_1y^2\\ f_2(x,y)=y+a_2x^2+2b_2xy+c_2y^2$ Determine a neighbourhood of $(0,0)$ as large as possible on ...
4
votes
2answers
316 views

Contraction Mapping question

Let X be the set of continuous real valued functions defined on $[0,\frac{1}{2}]$ with the metric $d(f,g):=\sup_{x\in[0,\frac{1}{2}]} |f(x)-g(x)|$. Define the map $\theta:X\rightarrow X$ such that ...