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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3
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2answers
617 views

Any good, undergraduate level introductions to Functional Analysis?

In my lower division math classes, my instructors referenced functional analysis as essentially the extension of linear algebra to infinite dimensional vector spaces along with some real analysis. As ...
2
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1answer
72 views

show that if $\displaystyle\lim_{n \to \infty} f(n+x)=0$ then $\displaystyle\lim_{x \to \infty}f(x)=0$ [duplicate]

Let $f : \left[0,\infty\right]\to \mathbb R$ be uniformly continuous. If $\displaystyle\lim_{n \to \infty} f(n+x)=0$ where $x$ is in $[0,1]$ then $\displaystyle\lim_{x \to \infty}f(x)=0$ ...
0
votes
1answer
21 views

Show that $f:A \to [a,b]$

Let $f_n \to f$ uniformly in $A$.When $f_n:A \to [a,b] , \forall n \in \mathbb{N}$,show that $f:A \to [a,b]$. That's what I have done so far: $$$$ $a \leq f_n(x) \leq b \Rightarrow lim_{n \to ...
1
vote
1answer
24 views

Integral's limit

Let $X$ be a Banach space and $A$ is a linear bounded operator on $X$. It is well known that for $|\lambda|> \|A\|,$ we have $$\|(\lambda I - A)^{-1}\| \leq \frac{1}{|\lambda|-\|A\|}.$$ Now, let ...
2
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1answer
21 views

how to prove translation identity: $(\mathcal{F}^{-1}m\mathcal{F}f)(x)= e^{ikx}[\mathcal{F}^{-1}m(\cdot+k)\mathcal{F}(e^{-iky}f(y))](x)$?

Let $m, f\in S(\mathbb R^{n})$=The Schwartz space. My question: How to prove: $$(\mathcal{F}^{-1}m\mathcal{F}f)(x)= e^{ikx}[\mathcal{F}^{-1}m(\cdot+k)\mathcal{F}(e^{-iky}f(y))](x);$$ where ...
1
vote
3answers
153 views

Convergence of $\sum\frac{1}{n^3-n^2}$

Does this infinite series converge or diverge? $$\sum\frac{1}{n^3-n^2} $$ I've tried every test I can think of but I can't figure it out. Is there a series greater than this that is convergent or a ...
0
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1answer
25 views

$\int_{|x|<t} |\mathcal{F}^{-1}f(x) |dx \leq C t^{\frac{n}{2}}\|f\|_{L^{2}}$?

Let $f\in L^{2}(\mathbb R^{n}).$ Fix $t>0,$ My Question:How to show, $\int_{|x|<t} |\mathcal{F}^{-1}f(x)| dx \leq C t^{\frac{n}{2}}\|f\|_{L^{2}}$ ? [We note $\mathcal{F}$ denotes the ...
1
vote
1answer
79 views

Imbedding theorem for Sobolev space $D^{k,p}$

I can't find any reference on the imbedding theorem for the Sobolev space $D^{k,p}$, which is defined by $$D^{k,p}(\Omega)=\{u\in L_{loc}^{1}(\Omega)|D^{k} u\in L^{p}(\Omega)\}.$$ Is it the same as ...
1
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2answers
29 views

upper and lower limits of sequences

Suppose that $t_n\leq s_n$ for all $n\geq N_0$, and $\{s_n\}$ converges to s. Prove that lim sup $t_n\leq s$. I want to somehow use the fact that lim inf $t_n\leq$ lim inf $s_n$ and lim sup $t_n ...
3
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1answer
252 views

Prove that $\mathbb{N}$ with cofinite topology is not path-connected space.

$\mathbb{N}$ is the set of natural numbers. Let $U_{\alpha \in A} \subset \mathbb{N}$ be the subset such that its complement $\mathbb{N}$ \ $U_\alpha$ is a finite subset. Then $T= \{\emptyset, ...
0
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2answers
21 views

Regularity of a function between two paraboloids tangents

I know that the regularity of a continuous function $u$ between two paraboloids tangents in a neighbourhood of a point $x_0$ is $C^{1,1}$. I'd like to see for example, how to prove that $u$ is ...
4
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1answer
82 views

incorrect proof of Hahn Banach Theorem

What is wrong with the following trivial proof of the Hahn Banach Theorem Hahn Banach Theorem: Let $V$ is a real normed vector space and $U$ a subspace. Then if $\phi : U \rightarrow \mathbb{R}$ is ...
0
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1answer
83 views

Interior ball condition in $C^2$ domains

Why a $C^2$ domain satisfies the interior ball condition? I accept a reference too. Thank you.
1
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2answers
79 views

Find region for which F(x,y) = (x+y)^2 is Lipschitz in y

As the title says, I need to find such a region. Taking any x, and any y1 and y2 I used the expression |F(x,y1) - F(x,y2)| and plugged in the function respectively for y1 and y2. Now I have to find ...
0
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1answer
39 views

Prove that $\mathbb{N}$ is not metrizable where $U$ is open if it is $U=\mathbb{N}$, $U = \emptyset$, or $\mathbb{N}$ \ $U$ is a finite subset.

$\mathbb{N}$ is the set of natural numbers. Any set $U$ is open if it is $U=\mathbb{N}$, $U = \emptyset$, or $\mathbb{N}$ \ $U$ is a finite subset. This defines a topology on $\mathbb{N}$. Prove ...
1
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1answer
56 views

inequality-why is it like that?

I saw the solution of an exercise and there it is used the following inequality: $$e^{-(n-1)x} \leq e^{-{(n-1)}} ,\forall x \in [0,+\infty)$$ Why is it like that? I haven't understood it.. $$$$ The ...
2
votes
1answer
63 views

Double integral and polar coordinates

Please, help me solve this double integral $$\int^{2\pi}_0d\varphi\int^{2}_1\frac{1}{\sqrt{\rho^3\cos^3\varphi+\rho^3\sin^3\varphi}}\rho\,d\rho$$ I really don't know how to figure out and carry of ...
1
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1answer
60 views

Double integral And polar coordinate system

I have to evaluate this integral over the domain D The Plot would be like this: I decided to use polar coordinate system using it It gives me this but I don't know the upper limit of ...
0
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1answer
29 views

Differentiability and basic definitions

If $f+g$ is differentiable at $a$, must $f$ and $g$ be differentiable at $a$? If " and $f$ is differentiable at $a$, must $g$ be differentiable at $a$? If $f*g$ is differentiable at $a$ and $f$ is ...
2
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1answer
38 views

Fast argument to see that the dual map of a projection is a projection

If $X$ is a Banach space and $U,V$ are closed subspaces, such that $X \cong U \oplus V$, then a continuous linear map $P:X \rightarrow X$ is called a projection if $P|_U =id$ and ker(P)=V. Now we ...
0
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2answers
47 views

why is the sequence of functions not continuous at each point?

Why is following sequence of functions discontinuous everywhere?? $$f_n(x)=\left\{\begin{matrix} f(x)-\frac{1}{n},x \in \mathbb{Q}\\ f(x)+\frac{1}{n}, x \notin \mathbb{Q} \end{matrix}\right.$$ where ...
1
vote
3answers
93 views

Expand a sin(x^3) in Maclaurin's series and find a 30th derivative at (0)

I have a big task and problems with it. I have to expand this function in Maclaurin's series. $$cos(x^{3})$$ I tried expand it as $$\sum_{n=0}^\infty (x^n)/n!$$ but it's for $cos(x)$. So i don't know ...
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0answers
47 views

Proving the Rietz-Fischer Theorem for $p = \infty$

Rietz-Fischer Theorem: Let $E$ be a measurable set and $1 \le p \le \infty$. Then every rapidly Cauchy sequence in $L^p(E)$ converges both with respect to the $p$-norm and pointwise almost everyone ...
0
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1answer
64 views

Absolute and conditional convergence of a series with $\sin(x)$

I have to explore absolute and conditional convergence of this function series I tried to find $a(n)$ and $a(n+1)$ terms of the series and then divide it and take a limit. But I've got nothing. ...
1
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1answer
127 views

weakly convergent subsequence implies strongly convergent

Statement: Let $X$ be a Banach space If $x_n \rightarrow x$ weakly and every subsequence of $\{x_n\}$ has a strongly convergent subsequence, then $x_n\rightarrow x$ strongly in $X$ Attempt: ?
0
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2answers
42 views

Is it true that $x^2-y^2=0$ iff $(x-y)(x+y)=0$

since it is biconditional, what i did was see if A->B is true and B->A is true. so for $x^2-y^2=0 \implies (x-y)(x+y) = 0$, the left hand side reduces to $x=y$. then i plugged into $(x-y)(x+y) = 0$, ...
0
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1answer
54 views

A question about $C^2$ domain.

Let $\Omega$ be a $C^2$ domain and assume that $0 \in \partial \Omega$ and that $e_n$ is orthogonal to the boundary of $\Omega$ at $0$. Then in a neighbourhood of $0$, we can put \begin{equation} ...
1
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2answers
86 views

Limit and integral properties of a continuous function

Let $f$ be a continuous function on $[0,\infty)$ such that $\displaystyle\lim_{x \to \infty}f(x)= c$. Show that $\displaystyle\lim_{x \to \infty} \frac{1}{x}\int_0^x f(s)\;ds = c$. I've tried ...
0
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1answer
34 views

Сhange the order of integration in the double integral

I have to change the order of integration in this double integral I've decided to divide it in two similar areas D1 and D2 And I've got the following result Can You chech it and state my ...
2
votes
1answer
98 views

An ellipse bigger than a circle

Suppose you have a unit ball in $B^2\subseteq \mathbb{R}^2$ and a point $A=(a,0)$ where $a>\sqrt{2}$. I would like to show there is an ellipse $E\subseteq\mathrm{conv}(B^2\cup\{A,-A\})$ such that ...
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0answers
23 views

Expand a function in Maclaurin's series

Please help me expand this function in Maclaurin's series to find an interval of convergence. I tried to turn it into this expression: but it doesn't give me the result. Help me please it's ...
1
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1answer
42 views

Is $l^p$ closed in $L^p$?

Let's assume we have a subspace $X$ of $L^p$ and we know that $X \cong l^p$(this should just mean isomorph no isometry is assumed here). Can we infer from this that $X$ is closed? I have just read a ...
0
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1answer
120 views

Holder's Inequality Proof Verification

Wikipedia outlines a nice proof of Holder's Inequality in the link provided. The fifth sentence in the proof reads: Dividing $f$  and $g$ by $\|f \|_p$ and $\|g\|_q$, respectively, we can assume ...
2
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0answers
35 views

Is all the nonelementary function that is not piecewise could be express as infinite series of elemetary function?

Are all the non-elementary functions that is not piecewise expressible as an infinite series of elementary functions? details about elementary function - ...
2
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2answers
82 views

Absolute and conditional convergence of function series

I have a problem. I have to explore absolute and conditional convergence of this function series Unfortunately. I didn't find in my reference any words about "absolute and conditional", Instead ...
5
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2answers
164 views

What are integrating factors, really?

I can follow the rationale for integrating factors well enough, but they still feel like voodoo to me. Every single description of integrating factors I've seen (and I've seen quite a few, including ...
0
votes
1answer
21 views

Evaluate an integral over $\mathbb{R}^{3}$ and Green's Theorem

Let $p(x_{1}, x_{2}, x_{3})$ be a smooth function in $\mathbb{R}^{3}$ decaying sufficiently rapid as $|x| \rightarrow \infty$. Why is $$\int_{\mathbb{R}^{3}}p_{x_{i}}\, dx = 0?$$ By the Gauss-Green ...
0
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2answers
47 views

Lipschitz does not imply fixed points

I have the following problem in mind: Let us say we have a function $f:X\rightarrow X$ (X is a complete metric space) and it respects that if $x\neq y$ then : $d(f(x),f(y))<d(x,y)$. My trouble ...
0
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1answer
47 views

Bounded functions are linear combinations of functions of absolute value one.

Is it true that any complex-valued bounded measurable function on a measurable space is a finite linear combination of functions of absolute value one? If true, how is this statement proved?
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2answers
41 views

Calculate ch(0.2) to the nearest 0.01

Help me calculate ch(0.2) to the nearest 0.01. I tried to rewrite ch as a series but I still don't know how to evaluate it and what to do with factorial Help me please. it's very important
1
vote
3answers
121 views

Find a radius of convergence of power series

I have to Find a radius of convergence of this power series I' ve decided to use D'alambert indication: Looking for a limit i meet a problem with a factorial Please. help me finish this ...
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2answers
372 views

explore the convergence of series with ln(n)

Help me explore the convergence of red color rounded series. On this photo (the equation below) I used radical indication but it doesn't show me the result. What would be better to use? ...
3
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1answer
68 views

Solve $\frac{1}{1^z}+\frac{1}{3^z}+\frac{1}{5^z}+\cdots=\frac{1}{2^z}+\frac{1}{4^z}+\frac{1}{6^z}+\cdots$ for $z\in \mathbb C$

My professor gave us this problem. Find all complex numbers $z\in \mathbb C$ such that $$\frac{1}{1^z}+\frac{1}{3^z}+\frac{1}{5^z}+\cdots=\frac{1}{2^z}+\frac{1}{4^z}+\frac{1}{6^z}+\cdots$$ I ...
0
votes
3answers
31 views

how can I show that $a^n\rightarrow 0$ if $0<a<1$

It is clear for the high school students, but I must use the 'elementary analysis' way to prove the statement. I tried to show it by assuming the contrary, but it didn't give me a good result.
0
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1answer
179 views

Prove that a pseudo-hyperbolic ball is a Euclidean ball. Find the radius and center of the Euclidean ball.

We have that the pseudo-hyperbolic metric in the open unit disk $\mathbb D$ is defined by $$ \rho(z,w) = |\phi_w(z)|, \qquad \phi_w(z) = \frac{w - z}{1 - \overline w z}$$ where $z,w \in \mathbb D.$ ...
0
votes
1answer
184 views

How do we find sub-sequences and limit points?

In my lecture notes I'm given the definition of a limit point as: A real number $a$ is called a limit point of a sequence $s_n$ where $n$ is a natural number, if there exists a subsequence $s_{n_k}$ ...
0
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0answers
62 views

Hermite interpolation with interior points

I am trying to solve the following problem: Given the conditions on a curve c(u) of degree 4 at the points -1, 0, 1 as: c(-1) = 4; c'(-1) = 4; c(0) = 6; c(1) = -4; c'(1) = -6; find the generalized ...
0
votes
1answer
30 views

How do i analyze this complex diagram?

I'm asking how to analyze diagrams like this : http://upload.wikimedia.org/wikipedia/commons/thumb/9/96/Complex_LogGamma.jpg/600px-Complex_LogGamma.jpg What do distinct colors here mean? What do the ...
0
votes
1answer
52 views

Power Series to solve non linear differential equations.

I've been revising Power series recently and their application when it comes to solving linear differential equations, but in this question I'm not sure what to do when it's a non linear function. I ...
1
vote
0answers
117 views

proving uniform convergence for a uniformly continuous function

I've been stuck on this question (2a) for about a day now! Question and havn't really gotten anywhere, I think that the series is that integral of a step function that approximates f ie: $\Gamma_n ...