Theoretical foundations of calculus: limits, convergence of sequences, construction of the real numbers, least upper bound property, and related analysis topics such as continuity, differentiation, and integration through the fundamental theorem of calculus.

learn more… | top users | synonyms

1
vote
2answers
36 views

Is there a relation eigenvectors and unitary operator.

I am trying the understand the spectral theorem as given in wikipedia link: https://en.wikipedia.org/wiki/Spectral_theorem I understand that eigenvectors are vectors and unitary operator is a ...
3
votes
1answer
63 views

Compact linear operator

Today in lecture we were told that for a linear compact operator $T$ on an infinite-dimensional Hilbert space with infinite-dimensional range, we have that $\ker(T)^{\perp}$ is infinite-dimensional, ...
1
vote
1answer
31 views

Range of operator always closed. Mistake in argument

Let $A \in L(X,Y)$ be a linear operator between Hilbert spaces and the operator $$\hat{A}: \ker(A)^{\perp} \rightarrow \operatorname{ran}(A)$$ is a restriction of $A$ which is bijective. Now ...
0
votes
1answer
108 views

If two Riemann integrable functions are equal except at one point, then the integrals are equal

Let $f(x)$ an $g(x)$ be integrable functions over $[a,b]$ and let $\alpha$ be a point of $[a,b]$ if $f(x) = g(x)$ for all $x\neq \alpha$, then $$\int_{[a,b]}f(x)dx=\int_{[a,b]}g(x)dx.$$ So is it okay ...
0
votes
2answers
33 views

If a positive term series is less or equal to a positive real number for any finite n, will $S_n$ still bound by the same number for $n \to +\infty$?

For example, if $S_n = \sum\limits_{k = 0}^n a_k$ $\leqslant$ $R$ for any finite positive integer $n$ where $R$ is a fixed real number, will $Sn$ still bound by $R$ for $n \to +\infty$? if so, how to ...
1
vote
1answer
81 views

Determine convergence of the series $\sum\limits_{n=2}^{\infty}\frac{\cos(\ln(\ln(n))}{\ln(n)}$

How to determine convergence of the series $\sum\limits_{n=2}^{\infty}\frac{\cos(\ln(\ln(n))}{\ln(n)}$ ? I tried to get somewhere with Integral criteria and with comparing to other series but ...
1
vote
0answers
37 views

Vector field from group action

Let $\Phi: G \times \mathbb{R}^4 \rightarrow \mathbb{R}^4$ be a group action where $G = \mathbb{R}/(2 \pi \mathbb{Z}).$ Then $$\Phi(\theta,(x_1,x_2,p_1,p_2)) = ( R(\theta) (x_1,x_2)^T, R(\theta) ...
3
votes
2answers
91 views

Is there a theory of transcendental functions?

Lately I've been interested in transcendental functions but as I tried to search for books or articles on the theory of transcendental functions, I only obtained irrelevant results (like calculus ...
0
votes
4answers
154 views

Find the radius of convergence of $\sum_{n=1}^{\infty}{n!x^{n!}}$

Find the radius of convergence of $\sum_{n=1}^{\infty}{n!x^{n!}}$. Should I look at this series as: $\sum_{n=1}^{\infty}({n!x^{(n-1)!})x^{n}}$? I am really confues here. In addition, any attempt to ...
20
votes
3answers
1k views

Does there exist a continuous function from [0,1] to R that has uncountably many local maxima?

Does there exist a continuous function from $[0,1]$ to $R$ that has uncountably many strict local maxima?
0
votes
0answers
19 views

simple performance calculation dependent on different factors

I have to decide performance of some exam, Performance is dependant on few factors: factor1 : x1 out of y1 where x1<=y1, e.g. I checked 6 options out of 10, nearer the value of x1 to y1, better ...
5
votes
4answers
238 views

Is it possible to work out the derivative of $e^x$ using the summation definition of $e = \sum_n 1/n!$?

So I know this question is a bit obtuse because usually we define $e$ in terms of the $\lim_{n \to \infty} (1 + 1/n)^n$ definition, and then compute derivatives of $e^x$ from there appealing to the ...
1
vote
2answers
37 views

Express g's Fourier coefficients using f's ones, if $g(x)=f(x+c)$.

The Fourier coefficients are defined (in our course) as: $$\hat{f(n)}={1\over 2\pi}\int_{0}^{2\pi}{f(t)e^{-int}dt}$$ I am asked to express g's coefficients as a combination of f's ones, given ...
2
votes
1answer
131 views

Can anyone prove D'Alembert Criterion (Dalambert) criterion for converging positive sequences?

This will most likely be on the exam, but it is not given in the text book. In my notebook I have this proof which I will type out, but it makes no sense. Here it goes: $$\text{D'Alembert ...
3
votes
1answer
40 views

Existence of a vector field which dominates the first local vector fields given by the charts of a locally finite covering

Let $M$ be a smooth manifold, let $\{U_i,\psi_i\}_{i\in I}$ be locally finite family of charts and let $K_i\subseteq U_i$ be compact subsets. Does there exist a vector field $X$ on $M$, such that ...
1
vote
1answer
107 views

Prove the concavity of the transformation from a concave function to another

Let's say we have $f_1$ and $f_2$, both strictly increasing and strictly concave on $[0,+\infty)$. $f_1(0)=f_2(0)=0$ and the difference $f_1-f_2$ is strictly positive and strictly increasing. That is, ...
9
votes
3answers
157 views

difficult problem in riemman integrals

Could anyone help me with the following problem? Because i have stuck. problem Let $f:[a,b]\rightarrow [0,\infty)$ be continuous and not the zero function. Prove that $$\lim_{n\to \infty} ...
2
votes
0answers
53 views

How to “uniformly” mollify a sequence without knowing the higher derivative.

The motivation and previous discussion of some related ideas can be seen here. My question: given a sequence $u_n\in u_0+H^1_0(\Omega)$ where $u_0\in H^1(\Omega)$ and $\Omega\subset \mathbb R^N$ is ...
1
vote
1answer
69 views

Prove that the reciprocal of a polynomial function $f(x)$ is uniformly continuous on $R$.

Prove that the reciprocal of a polynomial function $f(x)$ is uniformly continuous on $R$. (It is provided that the reciprocal of the function exists. In other words, $f(x)$ is never zero for any ...
3
votes
1answer
220 views

Why is '1' the multiplicative identity of complex numbers and quaternions?

I am not a mathematician. I studied electrical engineering. I encountered quaternions while trying to understand motion of mobile robots and how rotations are achieved. This question occurred to me ...
0
votes
0answers
32 views

Asymptotic approximation inside of an integral

Suppose we want to find $$ \lim_{n \rightarrow \infty} \int_{a}^b f_n(x) dx $$ for $a, b \in \mathbb{R}$, $a \le b$. Suppose there is a sequence of functions $g_n(x)$ such that $g_n(x) \sim f_n(x)$ ...
1
vote
5answers
62 views

Show that this limit is related to Euler number

I am calculating the limit $\lim_{n \rightarrow \infty} \left( \frac{n!^{\frac{1}{n}}}{n} \right)= \frac{1}{e}.$ I got this limit from wolframalpha, but don't know how to show this.wolframalpha
0
votes
1answer
26 views

Analysis Doubt on sequence and series of functions

I have seen in Rudin the following "if a compact class of bounded continuous functions on a compact metric space is not equi-continuous then that class contains a sequence which has no equi-continuous ...
4
votes
2answers
74 views

Mean value theorem for integration in two dimensions

The mean value theorem for integration says that, if $G$ is a continuous real-valued function defined over an interval, $G: [a,b] \to \mathbb{R}$, then the mean value of G on the interval is achieved ...
0
votes
1answer
51 views

There are two periodic functions $f(x)$ and $g(x)$, provide an example when $f(x)*g(x)$ is unbounded, and $f(x)+g(x)=0$ has infinitely many solutions

There are two periodic functions $f(x)$ and $g(x)$ which are defined on $\mathbb{R}$, provide an example when $f(x)\cdot g(x)$ is unbounded, and $f(x)+g(x)=0$ has infinitely many solutions ?
2
votes
1answer
143 views

Real analysis : Preliminary topics for - Measure Theory, Integration Theory, Differentiation and Integration [closed]

I have following syllabus to study in Real Analysis Subject. I want to know, What are necessary topics that I have to cover as a prerequisite for below syllabus. Actually I am unable to get direction ...
0
votes
1answer
19 views

Trying to show a set equality regarding sigma algebras

Let $X_{\alpha}$, $\alpha \in A $ be a collection. Let $ X = \prod_{\alpha} X_{\alpha} $. Suppose $\mathcal{M}_{\alpha} $ is $\sigma-$algebra on $X_{\alpha} $. The product sigma algebra on $X$ is ...
2
votes
3answers
38 views

Limit in the set of real sequences.

I have troubles trying to prove the following proposition: Let $S$ be the set of real sequences with ...
0
votes
0answers
60 views

Multiple integrals of dirac delta

I'm working on a problem where integrals of this form arise: $$ \int\limits_{x_1=-1}^1 \;\; \int\limits_{x_2 =-1/3} ^{1/3} \dots \int\limits_{x_n =-1/(2n+1)} ^{1/(2n+1)} ...
2
votes
2answers
48 views

limit superior of sequences in real line

Let $(a_n)$ be a bounded sequence in $\mathbb R$. Is the limit superior of $(a_{n_k})$ less than or equal to the limit superior of $(a_n)$ for any subsequence $(a_{n_k})$ of $(a_n)$?
0
votes
1answer
90 views

When the function is continuous, bounded of variations, absolutely continuous?

Let the function $f_a:[0,1] \to \Bbb R$ be defined by $$f_a(x)=\begin{cases} x^a \cdot \cos(\frac{1}{x}) & 0 < x \leq 1 ;\\ 0 & x=0.\end{cases}$$ Find all values $a\ge 0$ such that ...
1
vote
1answer
91 views

Integral of a nonnegative Lebesgue-measurable function on $ [0,1] $.

Let $ f $ be a nonnegative Lebesgue-measurable function on $ [0,1] $. Suppose that $ f $ is bounded above by $ 1 $ and that $ \displaystyle \int_{[0,1]} f = 1 $. Problem. Show that $ f(x) = 1 $ ...
2
votes
4answers
52 views

$f$ is integrable & continuous over $[a,b]$ , $\int_{a}^{b}f(x)dx \geq 0$ for any subinterval $(\alpha,\beta)$ of $(a,b)$, then $f \geq 0$ in $[a,b]$

Some known things about this problem are: if $f(c) < 0$, $a < c < b$, then $f(x) < f(c)/2$ in some neighborhood of $c$, but I am not exactly sure how to use this to get to my goal of ...
1
vote
2answers
43 views

If $f$ is integrable on $[a,\ b]$ and $\int_a^b f(x) \mathrm dx >1$, then there exists a point $c$ in $(a,\ b)$ such that $f(c) > \frac{1}{b-a}$

So far for this problem, to my understanding, for something to be integrable means that $U(p,\ f) - L(p,\ f) < \epsilon$ but not sure how exactly to move beyond there to show that there exists a ...
1
vote
1answer
86 views

Looking for Clarification on a proof of Density of Q in R

I am looking for some advice/help in regard to the proof that Q is dense in R, given in Walter Rudin's book "Principles of Mathematical Analysis". Mostly, I want to see if my reasoning is correct for ...
0
votes
0answers
23 views

How can i explain that: complete subspaces of R are separable implies that complete, unbounded subspaces of R contain unbounded sequences.

I know if a subspace of R is separable, thus exists a subset in that subspace that is enumerable dense, and what? I don't understand
0
votes
0answers
61 views

How to determine Taylor series expansion of function $f(x) = \frac{\cos(x)}{x}$ about $a=1$?

Given function is $f(x) = \frac{\cos(x)}{x}.$ $y = x - a , y = x - 1$. $x = y+1 , f(y) = \frac{\cos(y+1)}{y+1}$ How to get Taylor series expansion about $1$ of this function? If it was needed to ...
0
votes
0answers
41 views

If $\partial\Omega\in C^{2+\alpha}$ and $-\Delta\Theta=f\text{ in }\Omega$ with $f\in C_0^\infty(\Omega)$, then $\Theta\in C^{2+\alpha}$

Let $\Omega\subseteq\mathbb{R}^n$ be a bounded domain with $\partial\Omega\in C^{2+\alpha}$ for some $\alpha>0$ $f\in C_0^\infty(\Omega)$ $\Theta\in C^0(\overline{\Omega})\cap C^2(\Omega)$ be the ...
0
votes
1answer
54 views

compute $\nabla f$ for a function over a cone

Let $D$ be the cone $D=\{rt:r>0, t\in\Omega\}$ with $\Omega\subset S^{n-1}$. I want to show that $$ \frac1{r^2}\int_{B_r}\frac{|\nabla f(x)|^2}{|x|^{n-2}} dx= C(n,g)r^{2(a-1)} $$ where $C(n,g)$ is ...
0
votes
2answers
103 views

dominated convergence theorem application

Prove or disprove this statement: if $f_n : \mathbb R \to \mathbb R$ are integrable functions with $f_n \to 0$ pointwise and $|f_n(x)| \le \frac{1}{|x| + 1}$ for all $n$, $x$, then $\lim\limits_{n ...
0
votes
0answers
15 views

What conditions are needed for cpt support of anti-gradient?

I'm reading through a paper by Michael Christ, "On the $\bar{\partial}_b$ Equation for Three-Dimensional CR Manifolds" found in the Proceedings of Symposia in Pure Mathematics, Volume 52, Part 3. One ...
2
votes
3answers
188 views

Prove that set is bounded but has no max/min

Okay so I'm reading through a proof in my text book and I dont understand one of the steps. We are proving that the set $(0,2)$ is bounded and has neither a max or min. We start by knowing that $0$ ...
3
votes
1answer
49 views

Proof $x_n \to \inf (A)$

$A$ is a non empty set of $\mathbb{R}$ with $\inf (A) = m$ Proof there exist a sequence $a_n$ with $a_n \to m$ [edit: is there an x = inf(A) then we can chosse for all n x_n := inf(A) ] * We ...
1
vote
3answers
123 views

Taylor series expansion of function $f(x) = \frac{\arcsin(x)}{\sqrt{1-x^2}}$ [duplicate]

Determine the Taylor series expansion of function $f(x) = \frac{\arcsin(x)}{\sqrt{1-x^2}}$. $f(x) = \frac{\arcsin(x)}{\sqrt{1-x^2}} = \arcsin(x)\frac{1}{\sqrt{1-x^2}}$ It is known: (1.) ...
0
votes
0answers
49 views

Minimal boundary conditions for divergence theorem

I've noticed that some domain conditions of questions here were only supposed to be finite dimensional and bounded. And then the divergence theorem was applied in the answers. But if I'm not mistaken, ...
1
vote
1answer
56 views

An affine set $C$ contains every affine combinations of its points

Show that an affine set $C$ contains every affine combinations of its points. Proof by induction: From the definition of an affine set, we know that $\forall x_1,x_2\in C \text{ and } \theta_i\in ...
0
votes
2answers
58 views

How to find the Tangent space of a submanifold?

How can I determine the tangent space $T_pM$, $p\in M$, for the following submanifold? $$ M:= \{(x,y) \in \mathbb{R}^2|x^2-y=0\}\subset \mathbb{R},\quad p=(1,1) $$ I figured out, that I can ...
0
votes
1answer
23 views

How are the following inequalities concluded based on this first one?

$$I-\frac{\epsilon}{3} \leq s(f,T) \leq \underline{I} \leq \overline{I}\leq S(f,T) \leq I+ \frac{\epsilon}{3}$$ from this, the following is concluded, but how? $$1.\ \ \ 0 \leq |I-\underline{I}|\leq ...
4
votes
1answer
107 views

Group Action and Smooth Manifolds

I was wondering if it is sufficient for a compact (i.e. Hausdorff) smooth manifold $M$ to have a free group action of a finite group $G$ in order to conclude that $M/G$ is a compact smooth manifold? ...
0
votes
0answers
70 views

If $\lim_{x\to\infty}\left(f(x)+\int_{0}^xf(t)dt\right)$ exists, what about $\lim_{x\to\infty}f(x)$? [duplicate]

Given that $f(x)$ is continuous on $[0,\infty]$. If $\lim\limits_{x\to\infty}\left(f(x)+\int_{0}^xf(t)dt\right)$ exists then evaluate $\lim\limits_{x\to\infty}f(x)$