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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66 views

Everywhere continuous extension of a almost everywhere continuous function

Let $X$ be a locally compact Hausdorff space and $\mu$ a Radon measure. If $f$ is continuous outside a set $N$ of $\mu$-measure 0, does there exist an everywhere continuous $g$ such that $f = g$ on $X ...
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2answers
626 views

Geometrical Meaning of derivative of complex function

What's the geometrical meaning of $f'(z)$ in complex analysis, as we know in real analysis $f'(x)$ has meaning ie. Slope of curve or gives max/ min. But what does derivative $f'(z)$ has geometrical ...
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0answers
105 views

Differential Equation has a unique solution periodic

Let $A(t)$ continuous and periodic of period $S$ in $\mathbb{R}$. Suppose $x' = Ax$ has $\varphi \equiv 0$ as the only periodic solution of period $S$. Show that there exists $\delta> 0$ such that ...
3
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1answer
141 views

translation of open set problem

Suppose U is an open set in an Euclidean space. Then any point in U is contained in all but finitely many open sets that is translated by vectors converging to zero. It is easily proved in one ...
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1answer
46 views

Understanding the proof of $~M$ invariant set $\Rightarrow$ subtangential condition holds

Problem: I want to understand a proof of the claim given in the title. Suppose we have an initial value problem $\{\dot{x}=f(t,x)~,~x(t_0)=x_0\}$ with continious $f$ and solution $x(t)$. Proof: ...
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1answer
27 views

Least norm problem for any norm?

Now I have questions about the following variants of least square problem If $A$ is a $ m \times n$ matrix(which need not be full rank, but can assume $m \geq n$) and let $b \in \mathbb{R}^m$. Also ...
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1answer
19 views

Relation between $L^2(\mathbb{R}_+)$ and $L^1_w(\mathbb{R}_+)$

As an exercice, I'm looking to find an inclusion or equality relationship between $L^2(\mathbb{R}_+)$ and $L^1_w(\mathbb{R}_+)$ when $w: x \to x^{-1/2}$. Actually, I think that we have the inclusion ...
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1answer
92 views

The differential is NOT the Jacobi Matrix?

In the book Analysis II by C.T. Michaels the differential is introduced as the Jacobi-Matrix. In class we had the following definition: Definition: Let $U \subset \mathbb{R}^m$ be open, $f: U \to ...
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1answer
93 views

If $f \in C^\infty$, and $f$ is nonnegative and integrable in $\mathbb{R}$, can I say that $f^\prime$ is integrable?

I'm not sure how to describe the question any further in the title than it is, but I will try to explain what I have done. If $f$ is a Schwartz function, I believe that $f^\prime$ will always be ...
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1answer
113 views

Differentiable function strictly concave up $\iff f'$ strictly increasing

I feel like this is false, but I am stumped as to find a counter example. Would $f(x)=x^4$ be a candidate? Thanks!
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2answers
42 views

Existence of a differentiable path passing through 3 points in $R^{n}$.

Show that, given any three points of $R^{n}$, there is a differentiable path through these three points. I'm having difficulty solving this problem. I'm trying to solve this problem as follows: Let ...
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0answers
43 views

converge of a improper integral $\int_0^1\frac{dx}{\sqrt{x^3-x}}$

I am trying to show $\int_0^1\frac{dx}{\sqrt{x^3-x}}$ converge. But I cannot remember a suitable testing rule for this one. Could anyone help me? Thank you!
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2answers
75 views

differential forms, stokes theorem in higher dimension

The problem is: Consider the differential form $a=p_1dq_1+p_2dq_2-p_1p_2dt$ in the space of $R^5$ with coordinates $(p_1,p_2,q_1,q_2,t)$. (a) compute $da$ and $da\wedge da$ (b) Evaluate the integral ...
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1answer
47 views

Integral-Fourier sum

I am trying to prove the following relation in (3) where $\alpha,\beta,\gamma,\delta,\omega \in \mathbb{R}$. Given the integral $$ I=\frac{1}{2}\int_0^\alpha dx \left( \beta ...
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1answer
49 views

Show that a function $\psi : \Bbb R^n \to \Bbb R$ is affine

Fix a point x in $\Bbb R^n$. Let c be a point in $\Bbb R^n$ and define the function $\psi : \Bbb R^n \to \Bbb R$ by $$\psi(\mathbf u) = \langle \mathbf c, \mathbf u - \mathbf x \rangle \text{ for } ...
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3answers
123 views

Why $\log xy=\log x+\log y$?

It is of course well known and basic formula. I am just curious. Is there a proof for it? How to prove that $\log xy=\log x+\log y$?
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2answers
61 views

showing $\inf \sigma (T) \leq \mu \leq \sup \sigma (T)$, where $\mu \in V(T)$

I am trying to prove the following: Let $H$ be a Hilbert space, and $T\in B(H)$ be a self-adjoint operator. Then for all $\mu \in V(T)$, $\inf \left\{\lambda: \lambda \in \sigma (T) \right\}\leq \mu ...
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2answers
87 views

Cauchy sequences are bounded?!

I'm having trouble understanding the proof that Cauchy sequences are bounded, here's the proof I've been given Let $s_n$ be a Cauchy sequence. We take a concrete value of $\varepsilon$, for ...
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2answers
609 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 ...
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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 $\displaystyle\lim_{x \to \infty}f(x)=0$ ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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
249 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, ...
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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 ...
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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.
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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 ...
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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 ...
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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
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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. ...
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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: ?
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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 $(x-y)(x+y) = 0$, ...
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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} ...
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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
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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
vote
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 ...