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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2
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2answers
94 views

Dense subset of continuous functions

Let $C([0,1], \mathbb R)$ denote the space of real continuous functions at $[0,1]$, with the uniform norm. Is the set $H=\{ h:[0,1] \rightarrow \mathbb R : h(x)= \sum_{j=1}^n a_j e^{b_jx} , a_j,b_j \...
0
votes
1answer
32 views

Adding Sequences proof

Prove that if $a,b : \mathbb{N} \to \mathbb{R} $ are sequences with $\lim_{n \to \infty}{a_n} = L$ and $\lim_{n \to \infty}{b_n} = M$ then $\lim_{n \to \infty}{a_n+b_n} = L + M$ Prove that if $\lim_{...
0
votes
1answer
53 views

$How to determine if this series is convergent?

Let the sequence $\{a_n\}$ be defined as follows: $$a_n \colon= \begin{cases} \frac{1}{n^2} \mbox{ if $n$ is not the square of any positive integer}; \\ \frac{1}{\sqrt{n}} \mbox{ if $n$ is the ...
2
votes
3answers
35 views

Confused about limit proofs conceptually

In a question like this: Prove that if $\lim_{x \to a} f(x) = l$ and $\lim_{x \to a} g(x) = m$ then $\lim_{x \to a} \max(f(x), g(x)) = \max(l, m)$ In general, when asked for proofs like this, are ...
0
votes
1answer
272 views

Monotonic function; limits from the right and from the left

Can someone explain this to me? It seems quite easy, but somehow I can't manage to prove this on my own ... or in other words: when exactly does a function $f$ not have limits from the right and from ...
1
vote
1answer
31 views

Power series of $\sum_{n=0}^{\infty}\frac{a_{2n}}{2}z^{2n}$ and $\sum_{n=0}^{\infty}a_nz^n$

Consider the two complex power series $$\sum_{n=0}^{\infty}a_nz^n-(1)$$ $$\sum_{n=0}^{\infty}\frac{a_{2n}}{2}z^{2n}-(2)$$. Say the radius of convergence of (1) is R (finite). What can be said about ...
1
vote
1answer
31 views

$\text{limsup}|a_n|^{1/n}\geq \text{limsup}|a_{2n}|^{1/n} $

The question is very simple. Is it true that $\text{limsup}|a_n|^{1/n}\geq \text{limsup}|a_{2n}|^{1/n} $ where $<a_n>$ is some real sequence? Can this be proved or disproved? And also what if I ...
0
votes
2answers
65 views

proff of bounded integral [closed]

Let $f \colon [0,1] \to \mathbb R$ have a continuous derivative and $\int_{0} ^{1}f(x)dx=0$. Prove that $\forall \alpha \in (0,1)$ $$\left|\int_{0}^{\alpha}f(x)\, dx \right| \leq \frac{1}{8} \max \...
2
votes
1answer
71 views

if $f'(x)\rightarrow L$ as $ x \rightarrow \infty$, $-\infty \leq L \leq \infty $ then $ f(x)/x \rightarrow L $ as $x \rightarrow \infty$

If $f$ is differentiable on $(a,\infty)$, Show that if $f'(x)\rightarrow L$ as $ x \rightarrow \infty$, $-\infty \leq L \leq \infty $ then $ f(x)/x \rightarrow L $ as $x \rightarrow \infty$. ...
1
vote
0answers
47 views

Determining measurable sets

STATEMENT: Let $α$ be the non-decreasing function on $\mathbb R$ defined by $α(t) = 0$ if $t ≤ 0$ and $α(t) = 1$ if $t > 0$. Let $μ_α([a,b))=\alpha(b)-\alpha(a)$ , with $μ^*_α$ the corresponding ...
5
votes
1answer
97 views

Composition of relations. Both relations are functional and mutually inverse mappings. Zorich - MAI p22

$\def\R{\mathcal{R}}$ The composition $\mathcal{R}_2 \circ \mathcal{R}_1$ of the relations $\mathcal{R}_1$ and $\mathcal{R}_2$ is defined as follows: $$\mathcal{R}_2 \circ \mathcal{R}_1 := \{(...
1
vote
1answer
93 views

Complex power series - Radius of convergence

Let $f$ be analytic in the unit disk. Then we can write that as, $$f(z)=\sum_{n=0}^{\infty}\frac{f^{(n)}(0)}{n!}z^n,|z|<1$$ Now let $a_n=\frac{f^{(n)}(0)}{n!}$. So the radius of convergence $R$ is $...
0
votes
0answers
52 views

Uniform continuity of a continuous function

Let $g$ be a continuous function defined on the set $\mathcal{I}\times\mathcal{X}$. If $\mathcal{I}$ and $\mathcal{X}$ are compact sets, then how to prove the followings: $g$ is uniformly continuous ...
0
votes
1answer
74 views

Compactness and cartesian product

I'm having trouble figuring out how can I show that if two sets are compact then their cartesian product is also compact. Any help is much appreciated,thank you!
1
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2answers
120 views

Maximum and minimum function on an interval

Let $I := [a,b]$, where $a<b$. Suppose that $f$ is continuous and $1-1$ on $I$. Let $m$ denote the minimum value of $f$ on $I$ and let $M$ denote the maximum value of $f$ on $I$. (a) Carefully ...
4
votes
2answers
144 views

Stein simply connected slit

STATEMENT: Prove that the complex plane slit along the union of the rays $\cup_{k=1}^n\left\{A_k+iy: y\leq 0\right\}$ is simply connected. This is question 19 in chapter 8 of Stein's Complex Analysis ...
0
votes
1answer
42 views

Demonstrate the existence of the following limit

Prove that for $m \geq n \geq 1$ that $|a_m-a_n| \leq n^{-1}$ and deduce that $(a_n)$ converges. For $n\in \mathbb{N}$, denote $$a_n=\int\limits_1^n\frac{\cos(x)}{x^2}dx.$$ By integration by parts, ...
1
vote
2answers
54 views

convexity of $f(x)=x^{-a}e^x$

Consider the function $f(x)=x^{-a}e^x$ on $(0, \infty)$ with $a\in \mathbb{R}$. Study convexity. I know the derivative $f'(x)=(-a+x)x^{-a-1}e^x$ So if $a>0$ I can find the extremum points. but how ...
1
vote
2answers
26 views

How to adapt a profit function when given new variable cost

For a cost function I have been given the following information: fixed costs: 65\$ Production of 10 elements costs: 80\$ Production of 20 elements costs: 87\$ Producing 1 more when producing 10 ...
0
votes
1answer
21 views

Wrong simplification of a function?

I have 2 functions $f_1(x_1), f_2(x_2)$ and need to find the maximum of adding both functions, where $x_1 + x_2 = 50$. $f_1(x) = -0.001x^3+0.07x^2+5.9x-65$ $f_2(x) = -0.001x^3+0.07x^2+5.8x-20$ I ...
2
votes
0answers
43 views

An upper bounded for partial Fourier sum

Let $f$ be a Riemann integrable function on $[-\pi, \pi]$ such that $|\hat{f}(n)|\le \frac{K}{|n|}$ for some constant $K > 0$ and all $n\neq 0$. Show that $$|S_N(f)(x)|\le \sup_{y\in [-\pi, \pi]}|...
4
votes
1answer
145 views

Property of constant function

I learn real analysis and topology then I found something interesting about constant function. I am unsure it is true or false because I cannot prove it. I found property as follows: If $X$ is $T_1$ ...
1
vote
1answer
101 views

Asymptotic lines

I have a surface $f : \Omega \rightarrow \mathbb{R}^3$ that is represented by $$f(t, \phi) = (ae^t \cos(\phi),ae^t \sin(\phi), \int_0^t \sqrt{1-a^2 e^{2x}} dx)$$ I also calculated the matrix ...
1
vote
2answers
105 views

Bounded functions on a compact interval

If i have given $f:[0,1] \rightarrow \mathbb{R}$ $f$ is bounded. $g:[0,1] \rightarrow \mathbb{R}, x \rightarrow xf(x)$ And i have to prove $g$ continous in x=0. What can i say about $f$, is it ...
0
votes
0answers
46 views

Quasiconvexity of the composition of two functions

Consider $A: \mathbb{R}_{\geq 0}^n \rightarrow \mathbb{R}_{\geq 0}^{n \times n} $ and $B \in \mathbb{R}_{\geq 0}^{n \times m}$, and $c \in \mathbb{R}_{\geq 0}^n$. Assume that, for all $y \in \mathbb{...
1
vote
1answer
45 views

Limit law proof for max

I am working towards an extremely difficult real analysis problem. The statement is as follows: Prove that if $\lim_{x \to a} f(x) = l$ and $\lim_{x \to a} g(x) = m$ then $\lim_{x \to a} \max(f(x)...
0
votes
1answer
32 views

How to prove that $\sup\{a_nb_n|n\in N\}\le \sup\{a_n|n\in N\}\sup\{b_n|n\in N\}$

It is known that $$a_n,\ b_n\ge0.$$ And they are both upper bounded. Knowing this how can one prove that $$\sup\{a_nb_n|n\in N\}\le \sup\{a_n|n\in N\}\sup\{b_n|n\in N\}$$ I don't see how to approach ...
2
votes
1answer
693 views

Automorphisms of the upper half plane

STATEMENT: Suppose $(x_1,x_2,x_3)$ and $(y_1,y_2,y_3)$ are two pairs of three distinct points on the real axis with$$x_1<x_2<x_3 \;\;\;\;\text{and} \;\;\;\;\;y_1<y_2<y_3$$ Prove that ...
3
votes
1answer
33 views

Proof about outer measure. For an interval $I$, $|I|_e=v(I)$?

My question is when proving $|I|_e \ge v(I)$, why cannot I conclude from $S={I_k}_{k=1}^\infty$ is a cover of $I$, then $v(I)\le \sigma(S)$, so $v(I)\le inf \sigma(S)=|I|_e$? Why do we need the $I_k^*...
0
votes
0answers
32 views

Show that for a sequence $p_{n}$ of real numbers, $\limsup p_{n} < +∞$ iff $p_{n}$ is bounded above.

Show that for a sequence ${p_{n}}$ of real numbers, $\limsup {p_{n}} < +∞$ iff ${p_{n}}$ is bounded above. My Partial Proof: $\Leftarrow$ Given ${p_{n}}$ is bounded above, prove $\limsup {p_{n}} &...
2
votes
2answers
139 views

counter-example: aboslute convergence => convergence in incomplete vector space

Is the following statement true? Let $X$ be a normed linear space, $x_k \in X$, $k \in \mathbb{N}$ and $\sum_{k=0}^\infty \lVert x_k\rVert$ convergent. Then $\sum_{k=0}^\infty x_k$ is also convergent....
5
votes
1answer
64 views

How prove this limits is exsit $\displaystyle\lim_{n\to\infty}x_{n}$

let $f:[a,b]\to [a,b]$ be Continuous function,Assmue that sequence $\{x_{n}\}(n\ge 0)$ such $$x_{0}=x,x_{1}=f(x_{0}),x_{2}=f(x_{1}),\cdots,x_{n+1}=f(x_{n}),\forall n\in N^{+}$$ and $$\lim_{n\to\...
2
votes
1answer
41 views

higher moments of a r.v., combinatorical problem

I'm studying the book of Rick Durrett, I want to understand the proof of the Erdös Kac central limit theorem, so I also need to understand the Lindeberg-Feller theorem: for every $n \in \mathbb{N}$ ...
0
votes
2answers
117 views

corresponding system of equation of the given solution space

The following question seems to me interesting. it gives solution space and required the corresponding system of equation. The question is the following: Consider the vectors in $R^4$ defined by $...
2
votes
3answers
54 views

Strongly convergent to zero in $L^2$ but $H^1$ norm not vanishing

Let $\Omega$ be some open, bounded, smooth subset of $\mathbb{R}^n$. I'm wondering whether it is possible for a sequence of functions $f_n:\Omega \rightarrow \mathbb{R} $ to be strongly convergent to ...
0
votes
3answers
79 views

$\lim_{n\to\infty} \dfrac{a^n}{n!} = 0$ [duplicate]

Show that for any a in $\mathbb{R}$ $$\lim_{n\to ∞} \frac{a^n}{n!} = 0. $$ Hint: There exists a $n\in\mathbb{N}$ such that $n > |a|.$ I really do not know how to begin here with the proof and ...
5
votes
1answer
192 views

How can we estimate number of zeros?

Assume $a>0$ , $b>0$ and there exists a non-zero function $\phi(t)$ such that is the solution of $$y''+(a+b\cos 2t)y=0$$ and on $(-\pi/2,\pi/2)$ has $2n$ zero. How Floquet theory can help to ...
1
vote
1answer
75 views

Weak-* convergence in Sobolev spaces

Let's consider a sequence $\{f_n\}_n$ in the space $L^\infty(0,T;H^1(\mathbb{R}^n))$. What does it mean that $\{f_n\}_n$ converges weakly-* in $L^\infty(0,T;H^1(\mathbb{R}^n))$?
0
votes
2answers
80 views

Comput Spectrum of Idempotent

Let A be a unital banach algebra and a in A if a is idmepotent and a do not equal to 0 and 1 then the spectrum of a = {0,1}??
3
votes
1answer
37 views

Help with an Analysis problem involving isomorphisms and volumes.

I would like some help solving the following problem: Let $f: U \subseteq \mathbb{R}^{m} \rightarrow \mathbb{R}^{m}$ be a class $C^{1}$ function. Suppose that for some $a \in U$ the derivative of $f$ ...
1
vote
1answer
58 views

a question about compact set, how to prove there exits f(y)=y [duplicate]

Let (X,p) be a compact metric space.Suppose that f X->X is a function such that, for all $x_1$,$x_2$ $\in$X, if $x_1\neq x_2$ then p(f($x_1$),f($x_2$))<$p($x1$,$x2$)$. Prove that there exits a ...
1
vote
1answer
32 views

function of three variable is even than $f(a,b,c)=f(|a|,|b|,|c|)$

I used in the proof of Hlawka's Inequality you can find the link here Hlawka's Inequality that's if i have function of three variable is even in each variable, so that : $$f(a,b,c)=f(|a|,|b|,|c|)$$ ...
1
vote
1answer
31 views

Volume of a body bounded by planes

I'm just after the lecture about Fubini's theorem. And I "don't feel" how to do some exercises. Here is an example: What is the volume of the body bounded by: the graph of the function $f(x,y)=1+2x+...
0
votes
1answer
97 views

Integral of absolute value

I have the following integral which I want to make sure to solve correctly and transparently: \begin{equation} \int_{\mathbb{R}}\|e^{ax}\|dx \end{equation} If I take cases I obtain: $\int_{\mathbb{R}}\...
1
vote
1answer
39 views

sequence of close and bounded sets in a prefect space

Suppose that$(E_n)$$_{n \in \mathbb N}$ be a sequence of closed and bounded sets in complete space $M$ such that $ E_{n+1} \subseteq E_n$ for all $ n \in\mathbb N$. If $\lim \operatorname{diam} E_n $=...
0
votes
2answers
54 views

Can this inequality be solved with Mean value theorem

As my sub-assignment I have to solve inequality: $$ \ln\left(\frac{1}{x} + 1\right) -\frac{1}{x + 1} > 0 $$ If I understood MVT correctly, I should set $g(x)=\ln\left(\frac{1}{x} + 1\right) -\...
3
votes
3answers
153 views

An example of a function in $L^1[0,1]$ which is not in $L^p[0,1]$ for any $p>1$

Title says most of it. Could you help me find an example? It is easy obviously to show a function that would not be in $L^p[0,1]$ for a specific $p$ (say $(1/x)^{1/p}$, but I can't see how it would ...
0
votes
1answer
190 views

tangent space of manifold and Kernel

I want to understand the proof of this Theorem : Theorem: Let $ S $ be a submanifold of $ E $ and let $ a \in S $. Assume that there exists a submersion $f : E \supset U \rightarrow F$ of class $ \...
1
vote
1answer
34 views

\lim_{n\to ∞}c_n * a_n = 0

Let $(a_n)$ be a sequence in R that converge to 0 and $(c_n)$ be a bounded sequence. Show that $$\lim_{n\to ∞}c_n * a_n = 0$$. Obviously $\lim_{n\to ∞}c_n * a_n $ = $\lim_{n\to ∞}c_n * \lim_{n\to ∞}...
4
votes
5answers
200 views

How Prove this integral is diverge $\int_{0}^{1}\dfrac{dx}{\ln{x}\ln{(1-x)}}$

Show that this following integral is divergent (or diverges, if you prefer) $$\int_{0}^{1}\dfrac{dx}{\ln{x}\ln{(1-x)}}$$ I know when $x=0,1$ are singularities of the function and I want use this ...