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Questions tagged [real-analysis]

For questions about real analysis, a branch of mathematics dealing with limits, convergence of sequences, construction of the real numbers, least upper bound property and related analysis topics, such as continuity, differentiation, and integration through the Fundamental Theorem of Calculus.

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14 views

Proof of Plancherel Theorem

In Real analysis by Folland, the Plancherel Theorem is as follows: If $f \in L^1 \cap L^2$, then $\hat{f}\in L^2$, and $\mathcal{F}|(L^1 \cap L^2)$ extends uniquely to a unitary isomorphism on $L^2$. ...
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15 views

$Lip_\alpha$ is not closed in $C[0,1]$

As the title says I am trying to show that $Lip_\alpha$ is not closed in $C[0,1]$. $Lip_\alpha$ is the class of functions on [0,1] that belong to $Lip_\alpha([0,1];K)$ where $f \in Lip_\alpha([0,1];K)$...
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21 views

How to prove Ky Fan Inequality using Forward-Backward Induction?

The classical version of the inequality is: ${\frac {{\bigl (}\prod _{{i=1}}^{n}x_{i}{\bigr )}^{{1/n}}}{{\bigl (}\prod _{{i=1}}^{n}(1-x_{i}){\bigr )}^{{1/n}}}}\leq {\frac {{\frac 1n}\sum _{{i=1}}^{...
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1answer
18 views

Measure of the set of points with differential zero

Given a function $f$ which is defined and differentiable on interval $[0,1]$. Let $E=${$x\in [0,1]| f'(x)=0$}. Then the measure $m(E)$ would be zero. At first, I tried to prove this statement by ...
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1answer
43 views

Proving that $\lim_{(x,y)\to(0,0)} (x^2+y^2)^{x^2y^2}=1$ using polar coordinates

Am I doing this right? I rewrite the function as follows: $$(r^2\cos^2\theta+r^2\sin^2\theta)^{r^4\cos^2\theta\sin^2\theta} \stackrel{\text{various trig identities}}{=} r^{\frac{1}{4}r^4\sin^2 2\...
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0answers
9 views

About the sufficient conditions for an upperbound for an integral ratio.

Let f and g be functions, I am interested in finding (if there is any) the sufficient conditions on f and g such that we can satisfy an inequality of this type: $$\frac{\int_{0}^{\infty}f(x)dx}{\int_{...
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26 views

Showing $h_n$ does not uniformly converge

$$f_n(x)=x(1+1/n) \text{ if } x \in \mathbb{R}$$ $$g_n(x) = \begin{cases} (1/n)& x = 0, \text{ or } x \in \mathbb{I} \\ b+1/n& \text{if } x \in \mathbb{Q} \text{ with } ...
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3answers
37 views

Intuitively, $\sqrt{n}$ is not convergent. However, $|\sqrt{n+p}-\sqrt{n}|<\epsilon, \forall \epsilon>0, p\geq 1$

$|\sqrt{n+p}-\sqrt{n}|<\epsilon$ Clearly, $|\sqrt{n+p}-\sqrt{n}|=\frac{p}{\sqrt{n+p}+\sqrt{n}} \leq p/\sqrt{n} \rightarrow 0$. But by definition of a cauchy sequence, if we can choose $\exists N: ...
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26 views

Examples of linear forms on $\mathbb {R}^n$?

We discussed 1-forms in my analysis course today. I ws doing some background reading and keep seeing 1-forms refered to as a type of linear form. I see the definition but always find it helpful to get ...
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1answer
33 views

Choosing a scale $\beta>1$ to minimize the infinite sum $\sum_t (\beta t)^d \exp(-\beta^2 t^2)$

I am seeking to upper bound the following infinite summation, when $d$ is allowed to be arbitrarily large and $\beta>1$ is chosen, perhaps dependent on $d$, to minimize the sum: \begin{equation} f(\...
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0answers
27 views

Exponential decay of the gradient if the function itself and the Laplacian have exponential decay

Let $f: \mathbb{R}^n \rightarrow \mathbb{R}$ be twice continously differentiable. For $n=1$ one can prove using Taylors formula that we have $$ \sup_{\vert x \vert \leq R} \vert f'(x) \vert \leq 2\...
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1answer
31 views

Have a question about Open Mapping Theorem in functional analysis homework

Let X and Y be Banach spaces. Prove that T ∈ B(X, Y ) is surjective if and only if range(T) is not a meager subset of Y. I have no clue..hope somebody help me.. thanks!
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10 views

Power series as partial sums of coefficients.

Studying real analysis, there's an exercise, number 41 in chapter 10 in Elon Lages Lima's book "Um curso de análise vol. 1", that states: Let $(-r,r)$ be the convergence interval of $\sum a_nx^n$. ...
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3answers
38 views

Question about open interval with a finite decimal expansion

Let $Y$ denote the set of numbers in $(0, 1)$ with a decimal expansion that contains only $0$s and $1$s, and only finitely many $0$s. Decide if you think $Y$ is countably infinite or uncountable – I ...
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3answers
36 views

What is going wrong in the computation of the integral $ \int_0^{\pi} \frac{\sec ^2 x}{1 + \tan ^2 x} dx $?

If make the substitution $ u= \tan x$ then the limits of the integral both go to 0, giving us the wrong result of 0. My book suggests that this error is due to the discontinuity of the tangent ...
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1answer
27 views

Calculate initial velocity based on displacement, time and constant acceleration.

"A car has a constant speed along a road. It goes down a hill at a constant acceleration. 50s after it goes down the hill the speed is doubled and 50s later it reaches the end of the 200m hill and is ...
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0answers
26 views

Is there a solution to $(c+x)f'(c+x)=c(x-f(x))$?

(possibly under the condition f(0)=0). In general, is there a name for differential equations which feature the function f evaluated at another function of the argument, e.g. something like f(g(x))=f'...
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1answer
35 views

For any sequence $\{a_n\}$ has a subsequence $\{a_{n_k}\}$ such that $\lim \frac {a_{n_{k+1}}}{a_{n_k}}$ exists and belongs to $\{0, 1,∞\}$.

Let $\{a_n\}$ be a sequence of non-zero real numbers. Show that it has a subsequence $\{a_{n_k}\}$ such that $\lim \frac {a_{n_{k+1}}}{a_{n_k}}$ exists and belongs to $\{0, 1,∞\}$. If we consider the ...
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0answers
18 views

How to show there's a measurable set $K\subseteq[0,1]$ such that $\chi_K$ is everywhere discontinuous in $[0,1]\setminus N$ for any null set $N$?

I've been struggling with this homework problem for a while. I think I've found some constraints on $K$, but I have no idea how to create a set satisfying them. I think $K$ has to have: Both $K$ ...
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1answer
15 views

Coarea-like formula for BV function (not its derivative)

Let $f \in BV(\Omega)$. The coarea formula states that $$Df = \int_{\mathbb R} D \chi_{\{f >h\}} \, dh.$$ Do we also have that $$f = \int_{\mathbb R} \chi_{\{f >h\}} \, dh$$ holds?
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23 views

Proof verification. Show that $\mathcal{O}(\mathcal{O}(f)) = \mathcal{O}(f)$

Prove that: $$ \mathcal{O}(\mathcal{O}(f)) = \mathcal{O}(f) $$ I've started with letting some $u \in \mathcal{O}(\mathcal{O}(f))$, then: $$ |u| \le k_1|v| $$ Where: $$ |v| \le k_2|f| $$ Also: $$ \...
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1answer
39 views

what the application of the gamma function? [on hold]

Can I get perfect confident sources for applications of the gamma function? Sits or books or notes ,i get some sources and find 'incomplete gamma and distribution of gamma"like \begin{eqnarray*} \...
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0answers
15 views

Prove if $f$ is Lipschitz solution of $x'(t)=f(x)$ is well defined for all $t\in\mathbb{R}$

That's how I started $\frac{dx}{dt}=f(x)$, so $\log(f(x))=t+c$ therefore taking exponent of both sides we get $f(x)=Ae^t$ now lipschitz gives us $$\exists_C :\forall _{t_1, t_2} \ |Ae^{t_1}-Ae^{t_2}|\...
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0answers
17 views

$L^p$-space on the circle, question about the definition

I am reading a book by E. Zehnder and I am confused about an $L^p$-space he is using. Here's what is written in the book: Start by considering integrable functions $f \in L^1(S^1)$ which are ...
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1answer
22 views

Total variation and Lipschitz continuity

Let $f:B_R(0) \subset \mathbb{R}^N \to \mathbb{R}$ be a $L$-Lipschitz continuous function. Is it true that the total variation $|Df|(B_R(0))$ is controlled by the Lipschitz constant $L$? How?
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15 views

Characterization of summable families

Could somenone help me with to prove the following result? If $C=\{\lambda\in{\Lambda}:x_\lambda\neq{0}\}$ is countable and for every bijective mapping $\tau:\mathbb{N}\longrightarrow{C}$ $\sum_{n\...
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1answer
23 views

What is a good estimate for $\sum_{t=0}^T \frac{1}{\gamma^t}\frac{1}{\sqrt{t + 1}}$, where $0 < \gamma < 1$?

Let $0 < \gamma < 1$ and $T$ be a "large" nonnegative integer. Question What is a good estimate (upper bound) for $\sum_{t=0}^T\dfrac{1}{\gamma^t}\dfrac{1}{\sqrt{t + 1}}$? In general, for $\...
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55 views

How could I evaluate this limit $\lim\limits_{n\to\infty} \prod\limits_{k=0}^{n-1}\left(2+\cos{\left(\frac{k\pi}{n}\right)}\right)^\frac{\pi}{n}$?

This is an exercise related to the application of definite integral (Mathematical Analysis). That is, transform this limit into an integral. I tried to use $y=\ln{x}$, only to get $\int\limits_0^\pi ...
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0answers
16 views

General Method to solve Power Sum Inequality

This is a general method to have power sum inequality . We work with $x_i> 1$ $n$ real numbers . We want to show this kind of inequality : Let $x_i> 1$ be $n$ real positive numbers and $...
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1answer
27 views

Proof that every sum of exponents can be represented as a polynomial. I am missing an inital idea.

$$ s_n(p)=\sum_{k=1}^n k^p $$ Show: For every $q \geq 1$ exist rational numbers $ a_{k,q} , 1 \leq k \leq q-1 $, such that $$ s_n(q)= \frac 1 {q+1} n^{q+1}+ \frac 1 2 n^q + \sum_{k=1}^{q-1} a_{k,q}...
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2answers
111 views

Is “$\lim\limits_{n \to \infty}f(x_0+\frac{1}{n})=l$” another way of expressing the right-sided limit?

Let $f:\mathbb{R} \to \mathbb{R}$. Can we say that $\lim\limits_{n \to \infty}f(x_0+\frac{1}{n})=l$ is another way of expressing the right-sided limit at $x_0$? I tried to use the definition of the ...
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2answers
17 views

Doubt about definition of continuity involving open ball.

I've seen a definition of continuity on the resolution of an excersice that left me a little bit sceptical. The definition stated that a function $f$ is continuous at a point $x$ if $\forall \epsilon&...
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0answers
24 views

Bijection between $[a,b)$ and $(a,b)$? [duplicate]

I know this question has been asked and answered before, but I am working on my own through an analysis textbook and just wanted to check if the following construction would be appropriate: Define $a|...
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1answer
11 views

outer measure on R some sort of continuity about Measure

$m^*(E)=q>0$,for any $c\in (0,q)$,there exist $E_0\subset E$,such that $m^*(E_0)=c$ $$m^*(E)=inf\{mG|E\subset G ,\text{G is open set}\}$$ I think if I can find a set A$\subset E$,and $m^*(E-A)=c$,...
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2answers
56 views

Solution of second ordinary equation

i have the following question. Let $\phi_1$ and $\phi_2$ fundamental system solutions on an interval $I$ for the second order equation $$ y''+a(x)y= 0. $$ Prove that there exists fundamental system ...
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1answer
28 views

$f_n\in L^1$ and $f_n \to f$ in a.e and $|f_n|\leq g$ where $g\in L^1$ then $f\in L^1$

This is not duplicate I had specific question Please Help me I want to prove Dominated convergence theorem: I understand the whole proof except for one part. Why $f_n\in L^1$ and $f_n \to f$ in a.e ...
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0answers
44 views

Does there exist a map on real numbers of order 3? [on hold]

What is a function $f:\mathbb{R}\rightarrow \mathbb{R}$ such that $f\circ f\circ f(x)=x$ for all $x\in \mathbb{R}$?
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1answer
38 views

How to express the sum of the series $\sum_{i=1}^\infty \frac{x^n}{n^2}$ as an integral? [on hold]

How to express the sum of the series $$\sum_{n=1}^\infty \frac{x^n}{n^2}$$ as an integral? Any ideas to help me start?
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1answer
17 views

Limit of a sequence multiply by a constant

If the sequence $s_n$ converges to $s$, then for all $k \in \mathbb{R}$ the sequence $ks_n$ converges to $ks$. Proof: We assume that $k \neq 0$ since the result is trivial for $k = 0$ . Let $\...
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1answer
25 views

Proving $\int_{I} f \geq 0$ for an integrable function that is positive at rationals

Let $I$ be a generalized rectangle in $\mathbb{R}^{n}$ and suppose $f :I \rightarrow \mathbb{R}$ is integrable. Suppose $f(x) \geq 0$ if $x$ is a point in $I$ with a rational component. Prove $\int_{\...
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2answers
17 views

$[0,1)$ is in both $G_{\delta}$ and $F_{\sigma}$

I know $G_{\delta}$ is the complement of $F_{\sigma}$ and it can be proved easily by by using the countable intersection of sets is the complement of the countable union of the complements of the sets....
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1answer
30 views

The breakdown of the solution to the ODE $y' = \sqrt{4-2y}$.

Suppose $y=y(x)$ be a real valued function such that $$ y(x) = \int_0^x \sqrt{4-2y(t)} dt. $$ By differentiating both side with respect to $x$, we get the ODE $$ y' = \sqrt{4-2y}, \quad y(0)=0. $$ ...
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0answers
14 views

$L^1$ estimate for difference between a certain function and its average

How can we estimate the difference $$\int_{\mathbb{R}^N} \left| \chi_{x_N \le F(x_1, \dots, x_{N-1})} - \int_{B_R(0)}\!\!\!\!\!\!\!\!\!\!\!\!\!\! - \ \ \ \chi_{x_N \le F(x_1, \dots, x_{N-1})} dx_{1,\...
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1answer
15 views

Validity of a statement in a proof (regarding continuity)

Does the equality $f(a,b)=(f(a),f(b)) $ hold? I doubt it. Taking $f(x)= | x|$ and $a=-5, b=6$ gives us $f(-5,6)= (5,6)$, but $0 \notin (5,6)$. This is indeed a very stupid question, but I am not ...
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1answer
79 views

Find function $f(x)$ satisfying $\int_{0}^{\infty} \frac{f(x)}{1+e^{nx}}dx=0$

I am looking for a non-trivial function $f(x)\in L_2(0,\infty)$ independent of the parameter $n$ (a natural number) satisfying the following integral equation: $$\displaystyle\int_{0}^{\infty} \frac{f(...
0
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1answer
36 views

Determining if the limit exists of the sequence

I am trying to determine whether the limit of the following sequence exists and if so, find the limit. $f$ is a positive continuous function of $[a,b]$. \begin{align*} \lim_{n\to\infty} \Big[\int_{a}...
2
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2answers
26 views

An isometric mapping $f:(X,d) \to (X, d)$ is an open mapping

In general, the result doesn't hold. For example $f: \mathbb{R} \to \mathbb{R}^2$ with $f(x)=(x,0)$ [Euclidian Norm]. Now my attempt to prove the restricted case goes as follows: Let $V \subset X$ ...
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1answer
54 views

Does $\lim_{y \to 0} \int_{\mathbb{R}^N} F(x,y) = 0$ force $\lim_{y \to 0}F(x,y) = 0$?

Suppose $$\lim_{y \to 0} \int_{\mathbb{R}^N} F(x,y) = 0,$$ where $F:\mathbb{R}^N \times \mathbb{R} \to \mathbb{R}$ is a measurable non-negative function. Can we conclude $$\lim_{y \to 0}F(x,y) = 0$$...
2
votes
2answers
77 views

Sequence such that $\lim\limits_{n\to \infty} \frac{x_1^2+x_2^2+…+x_n^2}{n}=0$

Let $(x_n) _{n\ge 1}$ be a sequence of real numbers such that $\lim\limits_{n\to \infty} \frac{x_1^2+x_2^2+...+x_n^2}{n}=0$. Prove that $\lim\limits_{n\to \infty} \frac{x_1+x_2+...+x_n}{n}=0$. I used ...
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2answers
67 views

Proving $\displaystyle{\lim _{n\to\infty}}\sqrt[n]{n} = 1$ given that $\inf \{\sqrt[n]{n}|n\in \mathbb{N}\} = 1$ [duplicate]

Here's my attempt at trying to prove it by definition: $$\exists \mathcal{E} > 0 | \exists N\in \mathbb{N}|\forall n > N:$$ $$|\sqrt[n]{n}-1|<\mathcal{E}$$ But from this point on, I'm not ...