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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46 views

Uniform continuity and differentiation.

For (a) I have said By MVT: $|f(x)-f(y)|\le K|x-y|$ Choose $\delta=\epsilon/K$ $|f(x)-f(y)|=|(f(x)-f(y))/(x-y)||x-y|=|f'(c)||x-y|\le K|x-y| \le K\delta=\epsilon$ For (b) I have said ...
2
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0answers
71 views

Human accessible mathematics : Objects defined by a finite number of steps

This is about the uncountable spaces like real numbers : When i was a student, I was proud to talk about them as if they were little toys I could play with. But ... I once realized that "most" of ...
2
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0answers
65 views

Weak convergence in $L^1$

Does anyone have a reference for the following statement or similar ones? Let $U$ be an open bounded set in $\mathbb R^n$ and let $f\in C^0(U\times S^1)$. Then the sequence $f_m (x):=f(x,mx_i)$ ...
2
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3answers
99 views

What does 'finite-valued' mean?

In Rudin, while defining the concept of 'pointwise bounded', it says: if there exists a finite-valuded function $\phi$ defined on $E$ such that $|f_{n}(x)|<\phi (x)$. Here, I am quite puzzled by ...
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1answer
20 views

A Question About the Elements of $\ell^2$

Let $\ell^2 = \{(z_n) : \sum |z_n|^2 < \infty\}$ where $(z_n) \subseteq \mathbb{C}$. I just read a proof that made use of the fact that $(z_n) \in \ell^2 \implies (z_n)$ is Cauchy and hence $(z_n) ...
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1answer
79 views

Basic analysis - sequence convergence

I'm taking a course entitled "Concepts in Real Analysis," and I'm feeling pretty dumb at the moment, because this is obviously quite elementary... The example in question shows $\lim_{n\to\infty} ...
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1answer
43 views

Root test related: show $\lim_{n\to\infty}\sup_{k\ge n}|a_k|^{1/k}=\lim_{\infty}\sup_{k\ge n}|a_{k+1}|^{1/k}$

I've been trying to do the following proofs as an excercise. Show that $$\lim_{n\to\infty}\sup_{k\ge n}|a_k|^{1/k}=\lim_{n\to\infty}\sup_{k\ge n}|a_{k+1}|^{1/k}$$ This is what i have done so ...
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1answer
75 views

Continuous linear operator with noncontinuous inverse [duplicate]

This question gave an example of a continuous $f: E \rightarrow F$ which is bijective but has noncontinuous inverse. In the example, neither $E$ nor $F$ was a Banach space, are there any examples ...
2
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1answer
68 views

Question regarding $\epsilon-\delta-$proof

I want to prove the continuity of $f(x) = x^2$. Lets take $\epsilon > 0$ and $|x-x_0| <\delta$. I do: $$|f(x) - f(x_0) |= |x^2 - x_0^2| = |(x-x_0)(x+x_0)| < \delta |x-x_0|$$ Now the ...
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1answer
45 views

Prove something that is differentiable

The question states If g(x) is differentiable, then for any positive integer $n$, $(g(x))^n$ is differentiable and $\frac d{dx}$$(g(x))^n=(g(x))^{n-1}g'(x). $ Where does the continuity of g enter ...
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2answers
95 views

Suppose $x_n \ge 0$, $\sum_{n=1}^{\infty} \frac{x_n}{x_n+1}$ converges if and only if $\sum_{n=1}^{\infty} x_n$ converges.

One thing I notice we can do is say $\frac{x_n}{1+x_n} = 1 - \frac{1}{1+x_n}$ but I am not sure how this helps. By Cauchy criteria we can also say there exists $N \in \mathbb{N}$ such that ...
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1answer
20 views

Determine the largest open subset of C on which the functions below are analytic.

Determine the largest open subset of C on which the functions below are analytic. Give your reason. (i) $f(z)= \frac{2z+1} {z(z^{2}+1)}$ (ii) $f(z)= \frac{z^{3}+i} {z^{2}-3z+2}$ So I am brushing up ...
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6answers
83 views

convergence prove of $a(n) = \frac{n}{4^n}$

i need to prove that the following sequence converges: $$a(n) = \frac{n}{4^n}$$ in the assignment there is also a hint: prove that $2^n \gt n $ holds true for every $n \ge 0 $ i can prove that ...
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1answer
83 views

Prove $\log (1+x) = \sum_{n=1}^\infty (-1)^{n+1}\frac{x^n}{n}$

Prove $\log (1+x) = \sum_{n=1}^\infty (-1)^{n+1}\frac{x^n}{n}$, $\forall x \in (-1,1]$ This question is supposed to be done using Taylor's theorem (not the one involving integrals): For $ 0 < x ...
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1answer
100 views

Is the gradient operator surjective?

Let $\Omega \subset \mathbb{R}^{n}$ be open and bounded with Lipschitz boundary. Is the gradient operator $\nabla :H^{1} ( \Omega ) \rightarrow L^{2} ( \Omega )$ surjective? Here $H^{1} ( \Omega ) ...
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1answer
48 views

Properties of logarithmic mean.

I have been studying the logarithmic mean for the last few days now. Could someone please help me with the following two questions? 1) We know that the log mean is in between the geometric mean and ...
6
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2answers
126 views

Prove $ \left |\sin(x) - x + \frac{x^3}{3!} \right | < \frac{4}{15}$

Prove $ \left |\sin(x) - x + \dfrac{x^3}{3!} \right | < \dfrac{4}{15}$ $\forall x \in [-2,2]$ By Maclaurin's formula and Lagrange's remainder we have $\sin(x) = x - \dfrac{x^3}{3!} + ...
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1answer
68 views

Why the following is a seminorm rather than a norm

I really don't understand why the following is a seminorm rather than a norm? $$ p_k(u)=\sum_{|α|\le k}\sup_{x∈R^n}(1+|x|^2)^{k/2}|D^α u(x)|, $$ for all $u \in C^\infty$. I do understand if ...
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1answer
196 views

compact open topology on M(X,Y) continuous function space

I'm really confused with the statement. let $X$ be a set endowed with the discrete topology and let $Y$ be any topological space. Show that $M(X,Y)$ with the compact open topology is homeomorphic to ...
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1answer
315 views

Proof of convergence of an infinite product

a) Show that $\Pi_{n=1}^\infty x_n$ converges if and only if for all $\varepsilon>0$ there exists an $N$ such that for all $m\ge n\ge N$, $\left|x_nx_{n+1}\cdots ...
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1answer
94 views

The image of a Banach space under a continuous, linear, open map is a Banach space.

This is an exercise from Royden's Real Analysis. Suppose $X$ is a Banach space, there is a continuous, linear, open map from $X$ onto a normed linear space $Y$. Show that $Y$ is Banach.
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1answer
94 views

Topology of weak convergence

Edited: Thanks to etienne. I start with a compact metric space $(X,d)$. Then I consider the collection of finite measure $\mathcal{M}$ on $X$ and I equip $\mathcal{M}$ with the topology of weak ...
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1answer
28 views

About a sequence of continuous functions

Let $\Omega$ a bounded domain in $R^n$ . Let $u_k : \Omega \rightarrow R$ a sequence of nonnegative continuous functions with $u_{k+1} \geq u_k$. Fix $x \in \partial \Omega$ and supose that ...
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3answers
67 views

Lagrange Identity Proof

Was reading through Lagrange Identity Proof. However, one thing the proof assumes is $$\sum_{i=1}^p\sum_{j=1}^q a_i b_j=\sum_{i=1}^pa_i\sum_{j=1}^qb_j$$ which seems intuitive - but I wonder if ...
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0answers
31 views

Relation between two infinite products

Suppose $a_{n} \in \mathbb{C} - \{0\}$ and is such that $\sum 1/|a_{n}| < \infty$. Let $f(z) = \prod_{n = 1}^{\infty}(1 - z/a_{n})$ and $g(z) = \prod_{n = 1}^{\infty}(1 - z/|a_{n}|)$. For $0 < R ...
2
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1answer
38 views

Power series centered at $x =0$

I have this question in my advanced calculus textbook. Give an example of a power series centered at $x = 0$ that converges on $[-2,2]$ but not absolutely on the entire interval $[-2,2]$, and ...
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0answers
35 views

The property of positive fourier series. [duplicate]

This is the problem in the book 'Classical and multilinear harmonic analysis, volume 1' Let $f(x)=\sum_{n=0}^{N}[a_{n}\cos{2\pi nx}+b_{n}\sin{2\pi nx}]$ be a nonnegative function defiend on $[0,1]$. ...
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1answer
34 views

Is $f$ integrable in $L(X,\mathcal{X},\mu)$

Is $f$ integrable $L(X,\mathcal{X},\mu)$ $\mu(E)=\sum_{n\in E\cap\mathbb{N}} |n^2+n-6|$ $f:\mathbb{R}\rightarrow \mathbb{R_+}\cup\infty$ $f=(x-2)^{-4}$
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1answer
61 views

Proof check for $(X/M)^{*} \cong M^{\perp}$

I would like to know if the proof I have is correct. Statement: Let $M$ be a closed subspace if a Banach space $X$. Let $\pi: X \rightarrow X/M$ be the quotient map. Put $Y= X/M$ for each $\varphi \, ...
3
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1answer
265 views

Topologies of test functions and distributions

I'm wondering about some of the topological properties of $\mathcal D(\Omega)$ and $\mathcal D'(\Omega)$: I know $\mathcal D(\Omega)$ is not metrizable, so not first countable (right?). However, my ...
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1answer
1k views

Spectral Mapping Theorem

Spectral mapping theorem is as follows: https://math.uc.edu/~halpern/Matrix.methods/Homatrixmethods/Spectralmappingthm.pdf Is Spectral mapping theorem true for point spectrum ?
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1answer
22 views

Finding a function without knowing its structure but some conditions

I'm trying to find a function who meets this conditions but have no idea where to start. Just think it may be related to the function $Ca^{-\left(x-\mu\right)^2}$, If it really has this structure (or ...
0
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1answer
155 views

If the partial derivatives are continuous then the function is differentiable in the context of 3 dimensions

Context: This question has been bugging me for a while, mainly due to no knowledge of linear algebra and availability of only an ugly book Stewart's Calculus. There is a sufficient condition for a ...
1
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1answer
122 views

Laplacian(F) = (n-1/r)g'(r) + g''(r)

I got one more problem from my self reading of Methods of Advanced Calculus by Edwards, hints and solutions are equally appreciated: If f(x) = g(r), r= |x|, and n>=3, show that Laplace(f) = ...
3
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2answers
89 views

Is this a metric on R?

For $x,y \in \mathbb{R}$, define $d(x,y) = \sqrt{|x-y|}$. Is this a metric on $\mathbb{R}$? It's clear that $d(x,x) = 0$ and $d(x,y) = d(y,x)$ for all $x,y \in \mathbb{R}$. The triangle inequality ...
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0answers
63 views

The cardinality of the function

I'm reading a book about cardinality of functions and while I was solving some problems of the book I saw this: Prove that the cardinality of a general function $f:K \to K$ is $n^n$, where $n$ is the ...
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0answers
129 views

Use Taylor Theorem and Taylor Expansion to Prove

Let $g: R\rightarrow R$ be a twice differentiable function satisfying $g(0)=1, g'(0)=0$ and $ g''(x)-g(x)=0$, for all $x$ in R Fix $x$ in R. Show that there exists $M>0$ such that for all natural ...
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1answer
88 views

Serie of functions : interchange of limit of series

Let $\{f_n\}_{n=1}^\infty$ be a sequence of real-valued functions on $\mathbb{R}$. Show that if $f_n$ is continuous for all $n \in \mathbb{N}$ and the series $\sum_{n=1}^\infty f_n$ converges ...
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1answer
30 views

How to show that the limit of $\frac{\omega_1}{N^n}+\frac{\omega_2}{N^{n-1}}+…+\frac{\omega_n}{N}$ exist

$$\omega=(\omega_1, \omega_2,...)\in \{0, 1, 2,...,N-1\}^\mathbb N$$ How to show that the ...
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1answer
41 views

Solution to a general scaling problem $G(\lambda z)=\frac{G(z)}{\gamma z^n}$

When playing with the scaling problem $$G(4z)=\frac{G(z)}{2z}$$ (see also this question) I discovered, that the general problem $$G(\lambda z)=\frac{G(z)}{\gamma z}$$ with two constants ...
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0answers
38 views

Cardinality with property $x_1^2+x_2^2+…+x_n^2<1$

{$x_1,x_2,...,x_n$} of T (with no two of $x_1,x_2,...,x_n$ equal) has the property that $x_1^2+x_2^2+...+x_n^2<1$, then prove that T is a countable set. I do it in this way, in interval ...
2
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1answer
59 views

Question on a derivative on a Hilbert space

I have this functional $J(u)=\frac12 \|u\|^2+\int_0^1 F(t,Ku(t))dt$ where $F(t,u)=\int_0^u f(t,\xi) d\xi$,$\displaystyle Ku(t)=\int_0^1 G(t,s)u(s) ds$ with $G(t,s)=\begin{cases} s(1-t),&0\leq s ...
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2answers
40 views

Showing $ u =0 $ for a particular laplace equation.

Given open $\Omega\subset \mathbb{R}^n $ with smooth enough boundary, if for $x_0 \in \Omega$ there exists $ u \in C^2(\Omega\setminus\{x_0\}) $( we may as well take $u$ as smooth as we want) that ...
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3answers
1k views

Multiplication of infinite series

Why multiplication of finite sums $(\sum_{i=0}^n a_i)(\sum_{i=0}^n b_i)=\sum_{i=0}^n (\sum_{j=0}^ia_jb_{i-j})$ (EDIT: This assumption was shown to be false) does not work in infinite case? I have ...
0
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2answers
52 views

Quick question concerning the derivative of a power series

If we take $f(x) = \displaystyle \sum_{n=0}^\infty (-1)^n \dfrac{x^{2n+1}}{(2n+1)n!}$ wolframalpha gives the derivative of the function as $ \displaystyle \sum_{n=0}^\infty (-1)^n\dfrac{x^{2n}}{n!} = ...
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1answer
42 views

Rank Theorem question

Suppose that $f:\mathbb{R}^n\to \mathbb{R}^m$ is of class $C^1$ and $Df(x_0)$ has rank $m$. Then show there is a whole neighborhood of $f(x_0)$ lying in the image of $f$. My attempt: if $Df(x_0)$ is ...
0
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1answer
32 views

About an estimate of theorem 3 in Chapter 12 of Evans' book

This is the proof of theorem 3 in Chapter 12 of Evans' book as the following picture. I really don't understand why $|F(Du,u_t,u)|\le C(|Du|+|u_t|+|u|)$, because he didn't give us any restriction on ...
2
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2answers
35 views

What is this space with infinitely many different points with distance $1$ between any two different points?

I'm reading Mac Lane's: Mathematics, Form and Function: [...] There are also bizarre examples - such as "a space" with infinitely many different points, with distance $1$ between any two different ...
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1answer
50 views

On defining appropriate energy. Any principle?

I am reading Evans' book Partial differential equations. but I am really curious about how he define the appropriate energy? Is there any principle or rule to do this things? Because I notice that ...
0
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1answer
74 views

Sequence with lower bound on gaps [closed]

Suppose a sequence satisfies $|a_i-a_j| \geq 1/j$ whenever $i<j$. Can such a sequence be bounded?