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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1answer
71 views

Question on connected spaces and continuous functions

Let $f:X \rightarrow Y$ continuous where the metric space $(X,d)$ is conneced. On $Y$ we use the discrete metric. I want to show that $f$ must be constant on $X$. My approach: We may assume that $|X| ...
8
votes
3answers
182 views

Why does $\lim_{n \to \infty} \sqrt[n]{(-1)^n \cdot n^2 + 1} = 1$?

As the title suggests, I want to know as to why the following function converges to 1 for $n \to \infty$: $$ \lim_{n \to \infty} \sqrt[n]{(-1)^n \cdot n^2 + 1} = 1 $$ For even $n$'s only $n^2+1$ has ...
1
vote
1answer
45 views

$\frac{\partial Ψ}{\partial x}(x,y)=\frac{\partial Φ}{\partial y}(x,y)$ for all $(x,y)\in \Bbb R^2$

Give a piar functions $Φ:\Bbb R^2 \to \Bbb R $ and $Ψ: \Bbb R^2 \to \Bbb R$, it is often useful to known that there exists some contiunously differentiable function $f:\Bbb R^2 \to \Bbb R$ such that ...
0
votes
1answer
52 views

Question on derivative of a function

I have this exercise : We consider the système : $x_1'=x_2 , x_2'=-h_1(x_1)-x_2-h_2(x_3), x_3'=x_2-x_3$ ou $h_1$ et $h_2$ are locally lipschtizen , $h_i(0)=0$ and $yh_i(y)>0$ for all ...
1
vote
1answer
82 views

Potential function & constant

$\mathbf{Question:}$ Give a piar functions $Φ:\Bbb R^2 \to \Bbb R $ and $Ψ: \Bbb R^2 \to \Bbb R$, it is often useful to known that there exists some contiunously differentiable function $f:\Bbb R^2 ...
1
vote
3answers
111 views

I didn't understand the proof of the chain rule

From a lecture note about analysis: For differentiable functions $f$ and $g$ $$ (g\circ f)'(x_0)=g'(f(x_0))f'(x_0). $$ Proof: Consider the limit $$\lim_{x \to ...
2
votes
1answer
108 views

Please show me the question related to Continuously differentiable function & smallest value & minimizer

Suppose that the function $f: \Bbb R^n \to \Bbb R $ is continuously differentiable. Define $K= \{ x \in \Bbb R^n \mid \|x\|\le1\}$ a) prove that there is a point $x$ in $K$ at which the function $f: ...
4
votes
1answer
73 views

question about Lebesgue's integral

let $f(x)$ be a bounded measurable function defined on $\mathbb{R}$, then define $$F(x)=\int_0^xf(t)dt,\ \ x\in\mathbb{R}$$ We can see that $F(x)$ is a absolutely continuous function and by some ...
2
votes
2answers
244 views

Strong convergence of operators

I'm working through the functional analysis book by Milman, Eidelman, and Tsolomitis, and I have a question. The book states a lemma that I'm a bit confused about: A sequence of operators $T_n\in ...
3
votes
1answer
495 views

Conditions for Fubini's theorem

To preface this post, I have to admit that I have extremely little measure theory knowledge and I get lost when trying to read about Fubini's theorem for this reason. In the theorem statement for ...
0
votes
3answers
238 views

What does it exactly mean for a subspace to be dense?

My understanding of rationals being dense in real numbers: I know when we say the rationals are dense in real is because between any two rationals we can find a irrational number. In other words we ...
1
vote
1answer
70 views

Why this Equivalence of integrals is true?

$$\int_b^{\infty}(1+y^{2})e^{-y^2}\left[\frac{f(y)-f(x)}{y-x}\right]^{2}dy=O(1)\int_b^{\infty}e^{-y^2}\left[f^2(y)+f^2(x)\right]dy$$ enter link description here go to pg 72 in the end this pg. I ...
3
votes
2answers
225 views

locally finite open cover and $\sigma$-discreteness

The Definitions: A family $\{ A_s \}_{s \in S}$ of subsets of a topological space $X$ is locally finite if for every point $x \in X$ there exists a neighbourhood $U$ such that the set $\{ s \in S \mid ...
0
votes
1answer
153 views

Compactness and compact-finite measure in Lusin theorem (Rudin)

I have two questions about some hypotheses in Lusin's theorem as stated in Rudin's "Real and Complex Analysis". The proof initially deals with a subcase, that is the function $f$ is supposed to be ...
2
votes
2answers
423 views

Two notions of finer/coarser in Topology

As I learned in the beginning of Topology, if I have two topologies $\tau_1, \tau_2$ on a space $X$ such that $$ \tau_1 \subseteq \tau_2 $$ then $\tau_2$ is called finer then $\tau_1$ and $\tau_1$ is ...
1
vote
2answers
71 views

a question on orbit in ergodic theory

For the map $T: [0,1]\to [0, 1]$ defined by $Tx=10x\pmod{1}$, how to use the decimal expansion to construct a $x$, such that the orbit of $x$, say $\theta_x=\{T^nx: n\geq 0\}$ is dense in $[0, 1]$ ...
5
votes
1answer
164 views

Dirichlet Problem with piecewise smooth boundary

Suppose a domain $ \Omega \subset \mathbb{R^2} $ with $ \partial \Omega $. For $ f \in C^{\infty}(\mathbb{R^2}) $, the dirichlet problem is to find $ u $ with $ \Delta u = 0 $ in $ \Omega $, and $ f = ...
0
votes
3answers
235 views

Uniformly equivalent metrics and the metric on a countable product space

Two metrics $d_1, d_2$ on a set $X$ are called uniformly equivalent, iff for every $\varepsilon > 0$ there exists $\delta_1, \delta_2$ such that $$ d_1(x,y) < \delta_1 \Rightarrow d_2(x,y) < ...
2
votes
2answers
99 views

Find an equation of each plane tangent to $K$ which is parallel rto the plane $x-y+z=1$

Let $K$ be the cone given by $z=\sqrt {x^2+y^2}$ Find an equation of each plane tangent to $K$ which is parallel to the plane $x-y+z=1$ Sorry for not writing my ideas because I have No idea to ...
1
vote
1answer
171 views

convexity and lower semi-continuity for weak convergence

My question is a general one, whose answer can probably be found in any decent convex analysis book. I unfortunately don't have any at hand right now, so here it is: Let's consider a "reasonable" ...
0
votes
1answer
193 views

prove that $f(t)$ is orthogonal to $f'(t)$ for som all $t \in I$

Suppose that $I$ is nonempty open interval and that $f: I \to \Bbb R^m$ is differentiable on $I$ If $f(I) \subseteq \partial B_r(0) $ for some fixed $r>0$, prove that $f(t)$ is orthogonal to ...
3
votes
1answer
53 views

What is the meaning of $f(x) \rightarrow a$ as $g(x) \rightarrow b$?

The motivating example was the case: $$f(x, y)\rightarrow0\mathrm{\ \ as\ \ }\sqrt{x^2+y^2}\rightarrow\infty$$ What exactly does this mean? I might define it as: Any sequence $x_n$ with ...
0
votes
1answer
62 views

I solved the question. But I am asking a little bit. $\det(D(fog)(a))=?$

After here, how can I show its determinant?
0
votes
1answer
77 views

Convergence of translation operator

Set $T_t:L^2(\mathbb{R},dx)\rightarrow L^2(\mathbb{R},dx)$ the translation operator $(T_t(f))(x)=f(x+t)$. Is easy to show that $T_t$ is a continuous function and $||T_t||=1$ but I have to check if ...
2
votes
1answer
105 views

Laplacian Boundary Value Problem

Given a domain $ M \subset \mathbb{R}^2 $ and a function $ f : \partial M \rightarrow \mathbb{R} $, let $ \omega $ solve the boundary value problem: $$ \Delta \omega = 0 \text{ in } M \\ \omega = f ...
2
votes
1answer
245 views

Please explain me how can I show that the last limit does not exist?

I posted my answer with its question. But how can I show that the last limit -on the second page- does not exist? That is, $\mathbf{\lim_{(h_1, h_2)\to (0,0)}\frac{\sqrt {|h_1.h_2|}}{\sqrt ...
3
votes
1answer
115 views

Fourier series for $[x]-x+\frac{1}{2}$

$[x]-x+\frac{1}{2}$ has the Fourier series $$\sum_{n=1}^{\infty} \frac{\sin{2n\pi x}}{n\pi}.$$ By evaluating the series directly, which requires some work, it can be shown that the series is ...
2
votes
1answer
102 views

Name of the $(-1)^n$ function?

Does the function $f\left(n\right)=\left(-1\right)^n, n \in \mathbb{Z}$ used in a lot of mathematical formulas have a special name ? EDIT: The context of this question is that I need a name for this ...
3
votes
1answer
178 views

Small question about ODE

i have this question : Given three parameters $L,a$ et $\alpha$, we consider the differential equation : $$(E)\qquad x''+\alpha x' +a x + \sin x =L, \ > t\geq0$$ 1) Show that the ...
0
votes
1answer
87 views

Please only give me a feedback:) Verifying my solution - the differentiablity problem

$\mathbf{Question:}$ Let $r>0$, $f: B_r(0) \to \Bbb R$. Suppose there exists an $\alpha >1$ such that $|f(x)| \le \|x\|^{\alpha }$ for all $x \in B_r(0)$ (a) Prove that $f$ is ...
4
votes
0answers
118 views

Meaning of fractional Fourier transform with imaginary iteration count?

As one may know, the Fourier Transform $$F[f](\nu) = \int_{-\infty}^{\infty} f(t) e^{-2\pi i \nu t} dt$$ can be iterated, and this iteration generalized to fractional iteration count via ...
2
votes
2answers
194 views

Small sets on $\mathbb{R}$.

I was thinking of different definitions of small subsets on $\mathbb{R}$, such as meagre or zero-measure. These are quite well-known, so I was searching for different notions. Define a set has ...
0
votes
1answer
55 views

Showing a function is contractive

This seems to simple of a question and thus I am doubting myself... Show that the function $\dfrac{1}{2}x$ on $1\leq x \leq 5$ is contractive. \begin{align} |F(x) - F(y)| =& \left|\dfrac{1}{2}x ...
1
vote
1answer
353 views

Proof of the continuous function having tangent plane has directional derivatives

Suppose that the continuous function $f: \Bbb R^2 \to \Bbb R$ has a tangent plane at the point $(x_0, y_0, f(x_0, y_0))$ Prove that the function $f$ has directional derivatives in all directions at ...
0
votes
1answer
60 views

I know what I need to do but dont know how to apply: the question related to The first order approximation theorem

$\mathbf{Question:}$ Prove that $\displaystyle \lim_{(x,y)\to (0,0)} \dfrac{\sin(2x+2y)-2x-2y}{\sqrt{x^{2}+y^{2}}}=0$ $\mathbf{My\ ideas:}$ I will use the First Order Approximation Theorem. But ...
2
votes
1answer
47 views

Verifing the solution

$\mathbf{Question:}$ Let $f(x,y)=e^{\sin(x-y)}$ for $(x,y)\in \Bbb R^2$ Find the affine function that is a first order approximation to the function $f$ at the point $(0,0)$ $\mathbf{Answer:}$ ...
2
votes
1answer
78 views

Some statement about Cauchy product of sequences

Assume that we have two sequences $(a_n)_{n \in \mathbb Z}, (b_n)_{n \in \mathbb Z}$ such that for each $l\in \mathbb N $ the sequence $\left(|n|^l a_n\right)_{n \in \mathbb Z}$ is bounded, there ...
1
vote
1answer
108 views

Closed set in $l^1$ space

Let $$ X := \left \{ (a_n) : \sum_{n=0}^\infty |a_n| < \infty \right\}$$ with the metric $d(a_n,b_n) := \sum_n |a_n-b_n|$. Let $\delta_j^{(n)} := 1$ if $n = j$ and $0$ otherwise. Denote ...
0
votes
2answers
67 views

Which of these sets is a subspace of F?

Let $F = \mathbb{R}^\mathbb{N}$. I need to check which of these sets are subspaces of $F$: $F_1 := \{ x \in F:\ \text{$x$ is bounded}\}$, $F_2 := \{ x \in F:\ \text{$x$ is convergent}\}$, $F_3 := \{ ...
4
votes
1answer
41 views

For what kind of a subset its sums equal $\mathbb{R}^4$

For short, suppose $a,b$ are real numbers. Let $A=\{(\cos(at), \cos(bt), \sin(at), \sin(bt))\mid t\in \mathbb{R}\}$. Let $B=\sum A=\{\sum_{i=1}^n x_i\mid x_i\in A, n \geq 1\}$. For what values ...
6
votes
2answers
125 views

Prove that the series $\sum_{n=1}^\infty \left[f(n)-\int_n^{n+1}\!f(x)\,\text{d}x\right]$ converges

Let $f$ be a non-negative decreasing function on $[1,+\infty)$. Prove that the series $$\sum_{n=1}^\infty \left[f(n)-\int_n^{n+1}\!f(x)\,\text{d}x\right]$$ converges.
3
votes
3answers
512 views

$f$ integrable, $g$ measurable, $f = g$ almost everywhere implies $g$ integrable

If $f\in L(X,\mathcal{x},\mu)$, that is: $f\colon X\to R$ is measurable; $\int f^+\,d\mu<+\infty$ and $\int f^-\,d\mu<+\infty$; $\int f\,d\mu=\int f^+\,d\mu-\int f^-\,d\mu$. If $g\colon X\to ...
2
votes
2answers
3k views

The preimage of continuous function on a closed set is closed.

My proof is very different from my reference, hence I am wondering is I got this right? Apparently, $F$ is continuous, and the identity matrix is closed. Now we want to show that the preimage of ...
0
votes
1answer
96 views

Question related to partial differentiablity and directional derivative

$\mathbf {Question:}$ Define a function $f:\Bbb R^2 \to \Bbb R$ by $f(x,y)=$ $(x/|y|)\sqrt {x^2+y^2}$ if $y\not = 0$ $f(x,y)=0$ if $y=0$ $\mathbf{a)}$ prove that the function $f$ is not ...
1
vote
1answer
176 views

Checking my proof related to directional derivatives

Please can somebody check my answer? Tell me and explain me my mistakes and so on if there is. Thank you for helping :) Question: Suppose that the function $f:\Bbb R^n \to \Bbb R$ is continuously ...
1
vote
1answer
79 views
3
votes
1answer
580 views

Proof that a function with continuous partial derivatives has directional derivatives in all directions

I tried to prove it, but I would appreciate if someone could check my answer. I am just starting to learn real analysis on my own Thank you for helping. :) Theorem Let $f\colon \Bbb R^2 \to \Bbb R$ ...
1
vote
1answer
101 views

Question related to first order partial derivatives

If The funtion $f: \Bbb R^2 \to \Bbb R$ has directional derivatives in all directions at each point in $\Bbb R^2$ then the function $f$ has first order partail derivatives at each point in $\Bbb R^2$ ...
1
vote
1answer
67 views

a simple question about an inverse application

Bonjour to everybody. I have to explain some notations before asking a simple question quoted from my favorite exercise book. Sorry about that. First of all $\mathbb R$ is the set of real numbers. ...
3
votes
1answer
340 views

Set of all n-tuples is countable

I'm having trouble understanding the last part of the proof of this theorem (2.13) in the Rudin (blue) book: Let $A$ be a countable set, and let $B_n$ be the set of all $n$-tuples $(a_1,\ldots, ...