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
199 views

An example of a non-closable operator

I've encountered the following: Consider the usual Hilbert space $L^2([0,1],dx)$ and the dense subspace $\mathcal{D}=\mathcal{C}[0,1]$. Define $T$ on $\mathcal{D}$ by $T(f)=f(0)$. This is a ...
6
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2answers
418 views

Prove that if f(x) is integrable, then so is e^(f(x)).

So here is my question: I'm working on a homework problem that deals with Jensen's Inequality. It is a rather simple application, I believe, but I'm a little stuck. Here is the problem, along with ...
5
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4answers
99 views

Convergence of $\sum_{n=0}^{\infty}(-1)^n \frac{2+(-1)^n}{n+1}$

I have to show that the following series convergences: $$\sum_{n=0}^{\infty}(-1)^n \frac{2+(-1)^n}{n+1}$$ I have tried the following: The alternating series test cannot be applied, since ...
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2answers
153 views

Doubts related to a phase plane diagram.

I want to draw phase plane diagram of the following differential equation $$\frac{d^2y}{dx^2} - 2\frac{dy}{dx} + 10 y = 0.$$ Please check if my approach is correct. I have some doubts about it. ...
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2answers
106 views

Can any metric space be completed?

Completion defined in Real Analysis, Carothers, 1ed has been captured below. Can any metric space be completed?
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1answer
67 views

A question about multivariable concave functions

Consider a concave function $f: \mathbb{R}^N \rightarrow \mathbb{R}$. Is it possible for it to be convex in a single argument when I fix the remaining $N-1$ variables or does concavity of $f$ in $N$ ...
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1answer
154 views

Show that there is a sequence $(P_n)$ of partitions of $[a,b]$ such that $||P_n||\to0$ & $\lim_{n\to\infty} S(g,P_n)$ for the $g(x)$ defined.

Let $g(x)= \begin{cases} 0, & \text{if }x\in\mathbb{Q} \\ 1/x, & \text{if }x\not\in\mathbb{Q} \end{cases}$, $x\in[0,1]$. Show that $\exists$ sequence $(P_n)$ of tagged partitions of $[a,b]$ ...
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2answers
68 views

A Double Limit Question

Maybe it's easy, but: Is it true that $$\lim_{(x,y) \rightarrow (0,0)} \frac{x}{\sqrt{x^2+y^2}}=0$$ If it is, could you help me prove it? Thanks
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1answer
940 views

Must the intersection of connected sets be connected?

Must the intersection of two connected sets be connected? I believe the answer is no, but I am not entirely sure. I think a counter example would be a set that intersects another set in more than one ...
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1answer
146 views

Prove $\text{Beta}(x,y) = 2\int_0^{\pi/2}(\sin\theta)^{2x-1}(\cos\theta)^{2y-1}\,d\theta, \qquad \mathrm{Re}(x)>0,\ \mathrm{Re}(y)>0$

Prove that $$\int_0^1 x^{k}(1-x)^kdx=\frac{k!k!}{(2k+1)!}.$$ (Edit: Actually the proof can be found here http://en.wikipedia.org/wiki/Beta_function ) How would you show this $\text{Beta}(x,y) = ...
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3answers
81 views

$\varepsilon$-$\delta$ proof of $\sqrt{x+1}$

need to prove that $ \lim_{x\rightarrow 0 } \sqrt{1+x} = 1 $ proof of that is: need to find a delta such that $ 0 < |x-1| < \delta \Rightarrow 1-\epsilon < \sqrt{x+1} < \epsilon + 1 $ if ...
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1answer
79 views

Sequence with a contraction mapping of the sum

Consider a continuous function $f: \mathbb{R}^n \rightarrow \mathbb{R}^n$ with the following property. There exists $c \in (0,1)$ such that for all $x,y \in \mathbb{R}^n$ it holds that $\left\| f(x) - ...
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2answers
42 views

Proof inequality with log

How can I prove that there exist $n_0$, $c$ such that for all $n>n_0$: $$n^{\log_2{n}}\le c2^{n}$$ (So I mean the log of n with base 2). Can anybody help me?
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1answer
42 views

Is the following set connected?

Let $R \subset \mathbb{R}^2$ denote the unit square $R = [0,1] \times [0,1]$. If $F \subset R$ is finite, is $R \backslash F$ connected?
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3answers
71 views

Fast way to do this well-known integral (gaussian-distribution)

I want to evaluate $$ \frac{1}{\sqrt{2 \pi } \sigma}\int_{-\infty}^{\infty} x^2e^{-\frac{(x-\mu)^2}{2\sigma ^2}}dx.$$ The problem is, I don't want to run into heavy calculations. Therefore, maybe ...
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2answers
311 views

Uniform continuity on $[a,b]$ and $ [b,c]$ $\implies$ uniform continuity on $[a,c]$.

Let $f:\mathbb R \to \mathbb R$. Prove that if $f$ is uniformly continuous on $[a,b]$ and $[b,c]$, then $f$ is uniformly continuous on $[a,c]$. My attempt at a solution: I've came up with a solution ...
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1answer
73 views

How to show that h is constant by using **Liouville's theorem**

Let $h$ be an entire function. $\exists$ some $R \gt 0$ and $z_0\in \Bbb C$ s.t. open ball $B_R(z_0)$ isnt in the range of $h$. $B_R(z_0)\cap h(\Bbb C) \not = \emptyset{}{}$ How to show that h is ...
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1answer
122 views

Bounded Self-adjoint Operator on Hilbert Space

I am trying to show that if $A$ is a bounded, self-adjoint and positive operator on a Hilbert space $H$, $0 \in \rho(A)$, the following inequality holds for all $x \in H$ with $\|x\| = 1$: ...
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1answer
434 views

Proof that imaginary numbers exist? [duplicate]

How do imaginary numbers exist? I know you can't use the conventional number system, but use the complex one. But, how do you prove that the complex number system exists in the first place?
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1answer
375 views

Question from Folland real analysis 6.38

I have been staring at this for hours. I cannot figure out how to prove the following from Folland, problem 6.38. Show that: $$f \in L^p \iff \sum_{k=-\infty}^{+\infty}2^{kp}\lambda_f(2^k)<\infty$$ ...
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0answers
80 views

Is there a gap in Serre's proof of inverse function theorem?

On page 73 of 'Lie algebras and Lie groups', Serre proves the inverse function theorem for complete fields. I would like to have some clarification about the following point. Let $K$ be a complete ...
2
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1answer
59 views

Lattice points in spheres

Let $\mathbb{R}^n$ have the standard Euclidean metric and call a point $P = (x_1, \ldots,x_n)\in\mathbb{R}^n$ a lattice point if for all $i$, $x_i\in\mathbb{Z}$. Allowing small number theoretic ...
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1answer
42 views

tangent functions

let $A,B \subset \Bbb K $ $f:A \rightarrow \Bbb K $ $ g:B \rightarrow \Bbb K $ $ a \in \Bbb K $ We say f is tangent to g if $ a \in int (A \cap B)$ and $ \forall \epsilon > 0 \exists \delta ...
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2answers
134 views

Good Textbook on Topology

I have one year calculus and one year linear algebra background. In addition, I have had one semester study in metric space analysis. Can anyone suggest some good textbooks on topology, please? A ...
7
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1answer
144 views

Bound on $|f(x)|^2 + |f'(x)|^2$

Let $f\in C^2(\mathbb{R})$ be a twice differentiable function satisfying $$|f(x)|^2\le a$$ and $$|f'(x)|^2 + |f''(x)|^2\le b$$ for all real $x$, where $a$ and $b$ are positive constants. Prove that ...
3
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1answer
231 views

What are the benefits or losses of learning real analysis through a constructivist approach instead of a standard apporach?

Recently I've found some courses on real analysis that use the constructivist approach and I got curious on some aspects: What are the benefits of learning through this approach? Is it ok to learn ...
1
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1answer
61 views

product of the terms of two series that diverges to $\infty$

Suppose that $0\leq p_n\leq 1$ for each $n$. Also suppose $\sum_{n=1}^\infty p_n = \infty$ and $\sum_{n=1}^\infty (1-p_{n}) = \infty$. How can you prove $\sum_{n=1}^\infty p_n (1 - p_{n+1}) = \infty$? ...
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4answers
117 views

Help with a theorem about continuous increasing functions

I'm having a lot of trouble understanding the following theorem from my analysis class. Let $I$ be a subset of the real numbers and let $f : I \to R$ be increasing on $I$. Suppose that $c$ is an ...
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0answers
96 views

conjecture regarding the cosine fixed point

context/motivation if the angle on a calculator is set to radians, then it is very easy to demonstrate that iteration of $cos x$ (for arbitrary initial x) converges - simply keep pressing the ...
0
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1answer
51 views

epsilon delta approach to a problem

The following is what I would like to show. $$\lim_{x \to5}\frac{1}{x-3}=1/2$$ Given any $\epsilon>0$, $|\frac{1}{x-3}-\frac{1}{2}|\le\frac{|x-5|}{|x-3|}\le2\epsilon$ How do I get rid of ...
0
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1answer
797 views

Equality in Minkowski's theorem

I would like to see a proof of when equality holds in Minkowski's inequality. The proof is quite different for when $p=1$ and when $1<p<\infty$. Could someone provide a reference? Thanks!
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1answer
80 views

Is identity map one to one and onto?

Im reading a chapter of compactness in Real Analysis, Carothers, 1ed. Actually, identity map has been involved in and I've captured its definition: Equivalent Metrics As a last topic related to ...
2
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1answer
54 views

What is the limsup of $\sum \limits _{n=1}^\infty \frac{1+\sin{n}}{4}$?

$$\sum \limits _{n=1}^\infty \frac{1+\sin{n}}{4}$$ I computed the first $70$ terms of this series. They are each between $0$ and $1$, but they jump around quite a bit and I can't seem to determine ...
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2answers
73 views

Sinus equality proof [closed]

Show that the following equalities hold true for every $n$ from $\mathbb{N}$ and every $x$ from $\mathbb{R}$ $$\sin^{(n)}(x)=\sin(x+n\fracπ2)$$ How do I solve this?
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1answer
261 views

Proof: Legendre Polynomials Solving the Corresponding Differential Equation

In a homework question, we are asked to show that the Legendre polynomials do indeed solve the Legendre Differential Equation: According to Wikipedia, it is sufficient to show that after deriving ...
2
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1answer
85 views

Intro analysis - contraction mappings

A function $ f: \mathbb{R} \rightarrow \mathbb{R} $ is called a contraction mapping if there exists a positive constant K < 1 such that $ |f(x) - f(y)| \leq K |x-y| $ d) Suppose $f:\mathbb{R} ...
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1answer
31 views

set of linear and continuous functions

Let E,F be normed vector spaces $ \mathcal L(E,F) = \{ f \in Hom(E,F) | f-continuous\} $ Why 1) $Hom(\Bbb K^n,\Bbb K) = \mathcal L (\Bbb K^n,\Bbb K) = \Bbb K^n $ 2) $\mathcal L (\Bbb K^n, F) = F^n$
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1answer
131 views

integral of a continuous function bounded. Prove function is identically zero.

Let ${\rm f}:{\mathbb R}^{k} \to {\mathbb R}$ be a continuous function. Assume that for any $a > 0$ and any $k$-cell $Q_{a}$ of side length $a$ $\left(~\mbox{and therefore volume}\ a^{k}\right)$ we ...
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1answer
56 views

Lebesgue measure nested sequences

I am asked to prove the following: Let $E \subset \mathbb R$ be Lebesgue measurable. Then there is a sequence of open sets $(O_n)$ and a sequence of closed sets $(F_n)$ such that $F_n \subset E ...
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1answer
43 views

Convergence conditions of a series

Convergence conditions of: $\displaystyle\sum \frac{\sin(x^n)}{(x+1)^n}$ What i did: $\sum \frac{\sin(x^n)}{(x+1)^n} < \sum \frac{x^n}{(x+1)^n}$ And after I studied with the root test when ...
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1answer
66 views

Integrability of functions

Briefly justify the following facts: a) $|x|$ is integrable on $[-1,2]$ b) $x^{\frac{1}{4}}$ is integrable on $[0,9] $. c) The function $h(x)=\begin{cases} x^2& x\in[0,1] \\ ...
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1answer
67 views

Convergence of $\sum_{ k\in\mathbb{N} } \left( \frac{\lambda^k}{k!} \right)^n$

We know that $\sum_{ k\in\mathbb{N} } \frac{\lambda^k}{k!} = e^\lambda$. I'm interested in the convergence of $$S^{(n)}=\sum_{ k\in\mathbb{N} } \left( \frac{\lambda^k}{k!} \right)^n $$ for some value ...
2
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0answers
36 views

Characterizing Logarithm functions

Assume that $f:(0,\infty)\to\mathbb{R}$ is a differentiable monotone function satisfying $$f^{-1}(f'(x))=e^{1/x},\ \forall\ x\in (0,\infty)\tag{1}$$ If $f(x)=\log_a{x}$ for $a>0$ and $a\neq 1$ ...
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0answers
27 views

Self similar set and its measure

Prove: If $(K,\{f_i\}_{i=1}^N)$ is a self-similar set and $(\mu,\{\mu_i\}_{i=1}^N)$ is a self-similar measures, there is any arbitrary partition $\Lambda=\Lambda_a(r_1,\cdots,r_N)$ and ...
2
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1answer
50 views

analysis integral question from 0 to 1 inequality

When $f$ and $g$ being positive satisfy $f(x)g(x) \ge 1$ for every $x$ on $[0,1]$, then $$\int_0^1 f(x) dx \int_0^1 g(x) dx \ge 1$$ I can show this when $f$ and $g$ are monotone, but general ...
7
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1answer
326 views

Convergence of $\sum_\lambda \frac{1}{1-\lambda x}$ where $p(\lambda)=0$ for a certain polynomial $p$

The powers of the roots $\lambda$ of these polynomials $$p_n(x):=\sum_{k=1}^{n-1}\frac{n!}{(n-k)!}x^{k-1}$$ (compare with the $p_n$ here) sum to these values $$\sum_\lambda ...
5
votes
3answers
91 views

Does the series $\sum \limits _{n=1}^\infty \frac{1}{n} \left[{\frac{1}{(-1)^n-5}}\right]^{n}$ converge or diverge?

$$\sum \limits _{n=1}^\infty \frac{1}{n} \left[{\frac{1}{(-1)^n-5}}\right]^{n}$$ I applied the root test and believe that this series converges. That is, I found that: $$\text{limsup} ...
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0answers
15 views

$C_1$ Function Implies Continuous Directional Derivatives?

I'm wondering if a function being $C_1$ implies that the function has continuous directional derivatives. Thanks in advance!
0
votes
1answer
194 views

Convergence of events in a probability space with respect to $L^2$

Define for events $X, Y$ that $d(X,Y) = P((X-Y) \cup (Y-X)) $ = $ P(X \bigtriangleup Y) $, show that $d(X_n,X) \rightarrow 0$ if and only if $\chi_{X_n}$ converges in $L^2$ to $\chi_X$ (these are ...
1
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1answer
84 views

Continuous map on a compact metric space

Im reading a chapter of compactness in Real Analysis, Carothers, 1ed. Actually, I cannot understand a proof for uniformly continuity captured below: I cannot figure out a reason for the "Why?" ...