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
votes
2answers
75 views

A problem of constructing set to get finite summation

Given the function $f$, for any set $U$ with countable elements , define \begin{align} f(U)=\sum_{x \in U} \frac{1}{x} \end{align} Construct a set $A$, whose elements are positive integers, and ...
1
vote
3answers
412 views

supremum norm and submultiplicativity

If $f$, $g \in C(S)$ where $S$ is a compact set in $\mathbb{R}^n$ then it is true that $$\lVert fg \rVert \leq \lVert f \rVert \lVert g \rVert$$ where the norm is the usual supremum norm. Why is this ...
3
votes
1answer
362 views

Sobolev Spaces and Weak Derivatives

As you can probably guess, I'm currently studying about differential operators and functional analysis. We've studied the following theorem: A function $f \in L^2 (\Omega) $ lies in $ W^{1,2} ( ...
4
votes
1answer
65 views

Exercise about expressing grad f(x) in another basis.

I'm having difficulties with this exercise (from Elon LIMA's Curso de Análise, Vol. 2): $f:U\longrightarrow\mathbb{R}$ is function, differentiable on the open set $U\subset\mathbb{R}^n$. Let ...
1
vote
3answers
195 views

Solution of functional equation $f(x)=-f(x-a)$

I have a problem with finding solution. I suppose it will be something like $f(x) =G(x)\Re(e^{\frac{x\pi}{a}})$, where $\Re$ is real part of a complex number, $G(x)$ periodic function whith period ...
3
votes
2answers
285 views

Find a function that satisfies the following five conditions.

My task is to find a function $h:[-1,1] \to \mathbb{R}$ so that (i) $h(-1) = h(1) = 0$ (ii) $h$ is continuously differentiable on $[-1,1]$ (iii) $h$ is twice differentiable on $(-1,0) \cup (0,1)$ ...
0
votes
2answers
71 views

Intersection of $|z_1 - x|=r$ and $|z_2 - y|=r$

Let $x,y \in \mathbb{R}^k$ ($k\geq 3$), $|x-y|=d>0$ and $r>0$. Prove that if $2r>d$, then there are infinitely many $z\in \mathbb{R}^k$ such that $|z-x|=|z-y|=r$. Here's what I have ...
5
votes
1answer
323 views

Convergence of $\sum_{n=1}^{\infty }\frac{a_{n}}{1+na_{n}}$?

I need to prove the convergence/divergence of the series $\sum_{n=1}^{\infty }\frac{a_{n}}{1+na_{n}}$ based on the convergence/divergence of the series $\sum_{n=1}^{\infty }a_{n}$. It is given that ...
5
votes
2answers
1k views

Convergence of $a_{n}=\frac{1}{\sqrt{n}}\sum\limits_{k=1}^{n}\frac{1}{\sqrt{k}}$?

For $n$ in $\mathbb{N}$, consider the sequence $\left \{ a_{n} \right \}$ defined by: $$a_{n}=\frac{1}{\sqrt{n}}\sum_{k=1}^{n}\frac{1}{\sqrt{k}}$$ I would like to prove whether this sequence is ...
0
votes
0answers
76 views

description of a convex set of functions

I have a question about the characterization of a set of functions. Let $\phi$ a set containing all the functions $\phi(x): \mathbb{R}_{+}\mapsto \mathbb{R}_{+}$ that satisfy the following ...
7
votes
3answers
2k views

proving convergence of a sequence and then finding its limit

For every $n$ in $\mathbb{N}$, let: $$a_{n}=n\sum_{k=n}^{\infty }\frac{1}{k^{2}}$$ Show that the sequence $\left \{ a_{n} \right \}$ is convergent and then calculate its limit. To prove it is ...
3
votes
1answer
105 views

Evaluation of $L^p$ function

Functions in $L^p$ are only defined $µ$-almost everywhere, so for a given evaluation point $x$, $F(x)$, $f\in L^p$ can be changed to any value, so in general it would not be well-definied to just ...
1
vote
3answers
138 views

Can this continuous function be injective?

I've found very nice problem in math analysis book but I can't solve it: We define $c$ function as $c : \mathbb{I^2} \rightarrow \mathbb{R}$ and $\mathbb{I^2}$ means closed square in ...
0
votes
2answers
68 views

Distance in vector space

Suppose $k≧3$, $x,y \in \mathbb{R}^k$, $|x-y|=d>0$, and $r>0$. Then prove (i)If $2r > d$, there are infinitely many $z\in \mathbb{R}^k$ such that $|z-x| = |z-y| = r$ (ii)If $2r=d$, there is ...
0
votes
0answers
251 views

Schwarz inequality and linear dependence

Let $\{a_i\}$ and $\{b_i\}$ be families of complex numbers. I know that if $\{a_i\}$ and $\{b_i\}$ are linear dependence, then Schwarz inequality becomes equality, but I cannot prove the converse. ...
1
vote
1answer
54 views

Given $(a_n)$ and $(b_n) \in \mathbb{R}$,if we have $|a_n - b_n| < \frac{1}{n} \forall n \in \mathbb{N}$

Given $(a_n)$ and $(b_n) \in \mathbb{R}$,if we have $|a_n - b_n| < \frac{1}{n} \forall n \in \mathbb{N}$, I think it is possible to find an $N$ such that $\forall n \ge N$, we have $|a_n-b_n| < ...
0
votes
1answer
129 views

What is the meaning of this analysis problem and give some hint please?

What is the meaning of this analysis problem and give some hint please? This problem was founded on Analysis 1 by Herbert Amann and Joachim Escher on page 100. Determine the following subsets of ...
5
votes
2answers
162 views

Duals via a Bilinear map

Let $E$ and $F$ be normed vector spaces. Then if $B$ is a bounded bilinear form on $E \times F$ then every $y \in F$ defines a bounded linear functional $f_y$ where $f_y(x)=B(x, y) \forall x \in E$. ...
0
votes
1answer
42 views

show that integral converges even if it has a singularity

i am currently reading through a book on generalized functions, and there it is said that: ... $\int_{|x|\le r} |x|^{-t} dx$ converges for $t < n$ (in $n$ dimensions) and diverges for $t \ge ...
1
vote
2answers
56 views

Vectorial calculus statement proving

See this statement: $$ |u×v|^2=|u|^2\cdot|v|^2-(u\cdot v)^2 $$ I need to prove this is right. I only found that: $$ u×v=|u|\cdot|v|\cdot\sin\theta $$ and $$ u.v=|u|\cdot|v|\cdot\cos\theta $$ Does ...
1
vote
1answer
456 views

Proof of Riemann Integral of an indicator function of an interval on Real

The theorem is as follow: Let $a<b$ and let $c,d \in [a,b]$ with $c<d$. Then $1_{[c,d)}$ is Riemann Integrable over $[a,b]$ and $$\int_{a}^{b} 1_{[c,d)} dx = d-c$$ I am using Shroeder's ...
6
votes
2answers
360 views

Closure of the span in a Banach space

Let $X$ be a Banach space, and $S$ a subset. Is it true that $\overline {\operatorname{span}(S)}$ is equal to the set of the elements of $X$ that are obtained as norm convergent infinite sums of the ...
1
vote
1answer
178 views

Two questions about continuity of function between topological spaces

Let $X$ and $Y$ be topological spaces and suppose $f: X \to Y$ is continuous. If $f$ is continuous on $U \subset X$, will the restriction $f_U :U \to Y$ be continuous, if we consider $U$ to be a ...
2
votes
1answer
767 views

$f$ strictly increasing does not imply $f'>0$

We know that a function $f: [a,b] \to \mathbb{R}$ continuous on $[a,b]$ and differentiable on $(a,b)$, and if $f'>0 \mbox{ on} (a,b)$ , f is strictly increasing on $[a,b]$. Is there any ...
45
votes
6answers
3k views

Why don't analysts do category theory?

I'm a mathematics student in abstract algebra and algebraic geometry. Most of my books cover a great deal of category theory and it is an essential tool in understanding these two subjects. Recently, ...
0
votes
0answers
138 views

Showing map is Frechet differentiable (will add bounty)

Define the seminorm $$[u]_{\alpha} = \sup_{(x,t), (y,s) \in Q} \frac{|u(x,t) - u(y,s)|}{(|x-y|^2 + |t-s|)^{\frac{\alpha}{2}}},$$ and the norm $$\lVert u\rVert_{{C}^{k, \alpha}(\overline{Q})} = ...
2
votes
1answer
98 views

Matrix inversion of an analytical function

Following problem: I have a function $f(x_1,x_2)$ and Im looking for the inverse $finv(x_1,x_2)$ of the function which is defined through: $\int f(x_1,y)\cdot finv(y,x_2) d y =\delta(x_1,x_2) $ ...
5
votes
1answer
189 views

convergence of infimum

I have a question that I encountered during my internship: Consider a convergent sequence of continuous, convex functions $\{f_n(x)\}_n$ defined in $\mathbb{R}^M$. These functions are uniformly ...
0
votes
1answer
68 views

$A=\{x\in \mathbb{R}\mid b^x < y\}$ is nonempty

Let $1<b\in \mathbb{R}$ and $y\in \mathbb{R}$. I have proved that $A=\{x\in \mathbb{R}\mid b^x < y\}$ is nonempty when $y > 1$. Please give me any hint how to show that $A$ is nonempty when ...
2
votes
2answers
85 views

Question about $L^p$ spaces

Suppose $1<p<\infty$ and let $L^1$ and $L^p$ denote the usual Lebesgue spaces on $[0,1]$. Let $$A=\{f\in L^1:\|f\|_p\leq 1\}.$$ Show $A$ is closed in $L^1$. I took a sequence $\{f_n\}$ ...
0
votes
2answers
155 views

motivation of limit points

Lets use Wikipedia's definition of a limit point and let $\lim(A)$ denote the set of limit points of $A$. $a\in \lim (A) \leftrightarrow a\in\operatorname{cl}(A\setminus\{a\})$, $\lim (A)\cup A = ...
0
votes
1answer
65 views

Question regarding the function $R_X(t)=\frac{1}{\pi} \sum_{p\leq x} \frac{\sin(t\log p)}{\sqrt{p}}$

I want to show that the expected value $\mathbb{E}_{\omega ,T}(R_x(t)^{2k})$ behaves asymptotically as: $$\frac{(2k)!}{k!\cdot 2^k} \left(\frac{\log(\log T)}{2\pi^2}\right)^k$$ for $T^\epsilon < ...
3
votes
1answer
101 views

How to show that $\int \phi \,d\mu=-\int\phi'(x)f(x)\,dx$

Assume that $f\colon \Bbb R \rightarrow\Bbb R$ is left-continuous nondecreasing and let $\mu$ be a Borel measure in $\Bbb R$ such that $\mu([a,b))=f(b)-f(a)$ for $a<b$, $a,b \in\Bbb R$. I would ...
6
votes
2answers
95 views

Sharkovskii-type results in other topological spaces?

I recently came across Sharkovskii's Theorem which asserts that if $f:\mathbb{R} \to \mathbb{R}$ is continuous and has a cycle of length $m$, then $f$ has a cycle of length $n$ for any $n$ which comes ...
0
votes
1answer
54 views

If $ u \in W^{2,3} ( \Omega ) $ then $u \in L^3 ( \Omega )$?

If $ u \in W^{2,3} ( \Omega ) $ then $u \in L^3 ( \Omega )$ ? In wikipedia, the definition of Sobolev space is $$ W^{k,p} ( \Omega) = \{ u \in L^p ( \Omega)\mid D^{\alpha} u \in L^p , | \alpha| ...
4
votes
1answer
1k views

Why does $L^2$ convergence not imply almost sure convergence

What's wrong with this argument? Let $f_n$ be a sequence of functions such that $f_n \to f$ in $L^2(\Omega)$. This means $$\lVert f_n - f \rVert_{L^2(\Omega)} \to 0,$$ i.e., $$\int_\Omega(f_n - ...
9
votes
5answers
686 views

the sum: $\sum_{n=1}^\infty (-1)^{n+1}\frac{1}{n}=\ln(2)$ using Riemann Integral and other methods

I need to prove the following: $$1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\cdots+(-1)^{n+1}\frac{1}{n}+\cdots=\sum_{n=1}^\infty (-1)^{n+1}\frac{1}{n}=\ln(2)$$ Method 1:) The series $\sum_{n=1}^\infty ...
6
votes
1answer
205 views

Extending isometries between compact subspaces of Cantor space

Let $\omega$ be the set of natural numbers. $2^\omega$ is the Cantor space. Suppose $K$, $L \subset 2^\omega$ are compact, and there is an isometry $f: K \to L$. Then how could one extend $f$ to an ...
1
vote
2answers
1k views

set in $\mathbb{R}$ which is not a Borel-set [duplicate]

Possible Duplicate: Lebesgue measurable but not Borel measurable Constructing a subset not in $\mathcal{B}(\mathbb{R})$ explicitly if i start from the topology of $\mathbb{R}$, i.e. all ...
1
vote
1answer
66 views

Proving $\inf\limits_{f\in\Gamma} \{ F(y(t))(f(t))\}= -\|F(y(t))\| $

I would like to proof the next claim: Let $X$ a Banach space, $F\colon X\to X^*$ a linear continuous function, $$ \Gamma:=\{f\in (\mathcal{C}([0,1],X)\,:\, f(0)=f(1)=0\mbox{ and }\|f\|\leq 1\} $$ ...
0
votes
1answer
634 views

Question regarding the sequential characterization of limit on Reals

By the sequential characterization of definition of limit, suppose $f:D\to \mathbb{R}$, where $D\subset \mathbb{R}$, and let $a\in D$. If we want to show $\lim_{x\to a}f(x)= L$ by sequential ...
2
votes
3answers
257 views

Riemann Zeta formula

can anyone check if this formula is plausible ?? $$ \frac{1}{\zeta (s)} = \sum_{n=0}^{\infty}\frac{ (-\pi)^{n}(s-1)s}{2n!(s+2n)(s+2n+1)} $$ according to the authors this formula would be valid only ...
0
votes
0answers
552 views

Convolution between a kernel and an image with FFT

In the FFT2D paper (Fast Fourier transform used for a convolution with a kernel in the frequency domain), I'm lost at the second page first picture: ...
0
votes
3answers
718 views

Compact subsets of the real numbers

Let $C\subset \mathbb{R}$ be compact. I am wondering if $$C=\bigcup_{i=1}^n[a_i,b_i]$$ then for some $a_i,b_i\in\mathbb{R}$, $a_1\le b_1 < a_2 \le b_2 \dots < a_n \le b_n$. By Heine-Borel, ...
2
votes
1answer
138 views

A Question about limit of function on real

First of all, assuming ONLY the knowledge of sequential characterization of limit and also the epsilon-delta formulation of limit, why is the following limit undefined? $$\lim_{x\to 0} \sqrt{x}$$ ...
1
vote
1answer
109 views

excercise about implicit functions

Let $f(x,y)=\cos(x^2)+2xy+\sin(y^2)-4x-1+y$. Show that there exists an environment around zero so that $f(x,y)=0 \iff y=g(x)$ for a unique function $g$. Further show that $g$ is two times continuous ...
0
votes
2answers
120 views

proving inequality?

Here is another inequality I am trying to prove: Let $a,b,c$ be positive numbers. Prove that: $$1) \frac{bc}{a}+\frac{ac}{b}+\frac{ab}{c}\geqslant (a+b+c)$$ $$2) ...
0
votes
2answers
505 views

inequality using Lagrange Multipliers and Cauchy Schwarz inequality

Problem: I need to find the minimum of the expression: $$\sum_{k=1}^{n}a_{k}^{2}+\left(\sum_{k=1}^n a_k\right)^2$$ subject to the constraint: $$\sum_{k=1}^{n}p_{k}a_{k}=1$$ This problem can be ...
4
votes
1answer
220 views

Questions concerning a proof that $\mathcal{D}$ is dense in $\mathcal{S}$.

I am currently working through this lecture notes and on page 164, there it is said The space of $\mathcal{D}(\mathbb{R}^n)$ of smooth complex-valued functions with compact support is contained ...
1
vote
2answers
96 views

Solving $y'' - \frac{1}{x} y' + (1+\frac{\cot x}{x}) y = 0$ by rank reduction

With the substitution $$y(x) = \sin x \int u(x) \, dx\tag{*}$$ I managed to get to$$u'(x) = \left(\frac{1}{x}-2\cot x\right)u(x)$$ Solving which gave me $$u(x) = C_1 \frac{x}{\sin^2 x}$$ Inserting ...