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, integration through the fundamental theorem of calculus.

learn more… | top users | synonyms

1
vote
0answers
59 views

Is the repeating decimal $0.999… \in \,(0, 1)$? [duplicate]

Is the repeating decimal $0.999.... \in (0, 1)$? It seems like it can't be as $0.999...$ is defined as being equal to $1$.
1
vote
1answer
29 views

variation of a function over countable intervals

Let $f$ be a function of bounded variation on $[0,1]$. Let $\{[a_n,b_n]\}_{n=1}^\infty$ such that $(a_n,b_n)$ are pairwise disjoint and $\cup_{n=1}^\infty [a_n,b_n]=[0,1]$. (for example, $[1/2, 1], ...
1
vote
2answers
40 views

Intuition for sequences of functions?

A sequence $(a_n)$ of real numbers can be thought of as a function that maps $\mathbb{N}$ to $\mathbb{R}$. The supremum of this sequence, if it exists, will be some $k \in \mathbb{R}$. A regular ...
1
vote
1answer
41 views

The inner product determines the structure of the space

The Hilbert space employs inner product to determine the geometric structure,e.x. the angle. But I couldn't understand how. For example, the key structure of Euclidean space $\mathbb{R}^2$ is that it ...
1
vote
1answer
60 views

How do you self-study Functional Analysis?

It would be very handy to know about function spaces, distributions and Fourier stuff. It looks like Rudin's Functional Analysis covers these things, but I do not yet have the foundation for it. (see ...
0
votes
2answers
48 views

Maximum of $\frac{x(1-x)y(1-y)}{1-xy}$ over $[0,1] \times [0, 1]$?

I wish to find the maximum of $$\frac{x(1-x)y(1-y)}{1-xy}$$ over $[0,1] \times [0, 1]$, and where the maximum is achieved. I try to compute the gradient and set to zero, but this is not working out. ...
3
votes
1answer
99 views

Product of Lebesgue and counting measures

Let $\mathbb R$ be endowed with the standard Euclidean topology and let $\widetilde {\mathbb R}$ denote the line endowed with the discrete topology. Let $\mu$ and $\nu$ denote the Lebesgue and ...
3
votes
0answers
30 views

Let $(X,μ)$ be a measure space. Find a necessary and sufficient condition on $(X,μ)$ that $L_q(E) ⊂ L_p(E)$ for all $1 ≤ p < q ≤ ∞.$

Let $(X,μ)$ be a measure space. Find a necessary and sufficient condition on $(X,μ)$ that $L_q(E) ⊂ L_p(E)$ for all $1 ≤ p < q ≤ ∞.$ I want to say that the condition is that $E$ is finite. This ...
1
vote
0answers
28 views

Rapidly Decreasing Functions

Can someone explain the notion of a rapidly decreasing function? Namely, a function in the Schwartz space: $$\mathscr{S}(\mathbb{R}^n):= \{ f \in C^{\infty} (\mathbb{R}^n) : ||f||_{\alpha, \beta} ...
5
votes
1answer
87 views

sequence $a_n =\sum_{k=0}^{n-1}(-1)^k\frac{(n-k)^k}{k!}$ converge or diverge?

How check that sequence $a_n =\sum_{k=0}^{n-1}(-1)^k\frac{(n-k)^k}{k!}$ is converge or diverge ?
1
vote
2answers
71 views

$-\varepsilon\log(x)\overset{?}{\geq} -\log(\varepsilon x)$

I'm refering to this proof: http://en.wikipedia.org/wiki/Quantum_relative_entropy#The_result In there it's stated that "Since the matrix $(P_{ij})_{ij}$ is a doubly stochastic matrix and $-\log$ is a ...
-1
votes
2answers
28 views

Determine all values of $p \in R$ such that the sequence is in $l^2$. $\left\{\frac{1}{\sqrt{k}(\ln k)^p}\right\}_{k=2}^{\infty}$ [on hold]

Determine all values of $p \in R$ such that the sequence is in $l^2$ $$\left\{\frac{1}{\sqrt{k}(\ln k)^p}\right\}_{k=2}^{\infty}$$
-1
votes
1answer
29 views

derivative of a smooth compactly supported functions is also compactly supported [on hold]

I would want to know if the following implication is right: Define $ \mathcal{D}(\Omega):=\{\varphi(x)\ |\ \varphi \in C^\infty(\Omega) \text{ and } \varphi \text{ has compact support} \}$ $$ \phi(x) ...
-1
votes
0answers
21 views

Determine all values of $p \in R$ such that the sequence is in $l^2$. $\left\{\frac{k^p}{p^k}\right\}_{k=1}^{\infty}$ [on hold]

Determine all values of $p \in \mathbb{R}$ such that the sequence is in $l^2$ $$\left\{\frac{k^p}{p^k}\right\}_{k=1}^{\infty}$$ I'm so lost!
0
votes
0answers
38 views

Proving pointwise convergence of Fourier series [on hold]

I'm trying to prove the pointwise convergence of a $2\pi$-periodic function $f$ to its Fourier expansion. The proof on my lecture notes stops at this formula: $$f(t)-P_{N,f}=\frac 1 {2 \pi} ...
0
votes
2answers
34 views

Convergence of improper integral of $\ln f(x)$

Is there something know about the convergence of $\int_0^1 \ln f(x)dx $ for $f(x)$ continous on $\left(0,1\right)$ and both limits exists, i.e. $\lim_{x\to 0} f(x)$ and $\lim_{x\to 1} f(x)$ ? I ...
1
vote
1answer
24 views

If $p$ and $q$ are positive real numbers, show that $\sum_{k=2}^\infty(-1)^k\frac{(lnk)^p}{k^q}$ converges [on hold]

If $p$ and $q$ are positive real numbers, show that $$\sum_{k=2}^\infty(-1)^k\frac{(ln k)^p}{k^q}$$ converges
0
votes
1answer
22 views

Test for absolute convergence $\sum_{k=1}^\infty \frac{(-1)^{k+1}k^k}{(k+1)^k}$

Test for absolute and conditional convergence. $$\sum_{k=1}^\infty \frac{(-1)^{k+1}k^k}{(k+1)^k}$$ $$\lim_{k\to\infty}|a_n| = \lim_{k\to\infty} \frac{k^k}{(k+1)^k}$$ I'm stuck on what to do next.
2
votes
0answers
35 views

Show that $\int_E f (x, y) dx$ is differentiable with respect to $y$ and $\frac{d}{dy}\int_E f(x,y)dx=\int_E \frac{d}{dy}f(x,y)dx.$

Assume that $f = f(x,y)$ is a function defined on $E × (a,b).$ For each fixed $y ∈ (a,b),$ $f$ is integrable with respect to $x$ on $E$, and for each fixed $x ∈ E$, $f$ is differentiable with respect ...
0
votes
3answers
57 views

Proof that the continuous image of a compact set is compact [duplicate]

Let $X\subset \mathbb R^{n}$ be a compact set, and $f :\mathbb R^{n}\to \mathbb R $ a continuous function. Then, $F(K)$ is a compact set. See, I know that this question may be a duplicate, but the ...
8
votes
2answers
188 views

A Binet-like integral $\int_{0}^{1} \left(\frac{1}{\ln x} + \frac{1}{1-x} -\frac{1}{2} \right) \frac{x^s }{1-x}\mathrm{d}x$

I met this integral $$ \int_{0}^{1} \left(\frac{1}{\ln x} + \frac{1}{1-x} -\frac{1}{2} \right) \frac{ \mathrm{d}x}{1-x} \qquad (*) $$ while evaluating this log-cosine integral. I made several ...
1
vote
0answers
20 views

Showing a function is upper semicontinuous

Let $f: \mathbb{R} \rightarrow [0, B]$ and for every $\varepsilon > 0$, let $\varphi_{\varepsilon}(x) := \sup_{\{y: |x - y| < \varepsilon\}}f(y)$. Since for each fixed $x$, ...
1
vote
1answer
31 views

Proving and visualizing $\mathbf 1_{(x,x+a]}(y) = \mathbf 1_{[y-a,y)}(x)$

Here is a trick from one of the proofs in probability: $$\iint \mathbf 1_{(x,x+a]}(y) \ \lambda(dx) \ \mathbb P(dy) = \iint \mathbf 1_{[y-a,y)}(x) \ \lambda(dx) \ \mathbb P(dy)$$ for $a>0$. So ...
3
votes
1answer
35 views

Calculating limit in parts. Why possible?

Let $f$, continuous function, differentiable at $x=1$ and $f(1)>0$. Consider the following equation: $$\lim \limits_{x\to 1} ...
3
votes
1answer
119 views

Monotone Convergence Theorem for Riemann Integrable functions

I'm having a really hard time proving this statement (this is not homework): If $f_{n} : [0,1] \rightarrow \mathbb{R}$ is a Riemann integrable function for all $n \in \mathbb{N}$, and $0 \leq f_{n + ...
2
votes
1answer
41 views

Are these two expression square integrable?

I have two expressions (let's call them functions $f,g$) on $[0,1]$, where I want to find out whether they are square-integrable or better: for which $m \in \mathbb{Z}$ they are square-integrable ( ...
0
votes
0answers
31 views

Convergence of norms

I have this space $H_{0,p}^1=\lbrace u\in AC([0,+\infty),\mathbb{R}),u(0)=u(+\infty)=0, \sqrt{p} u'\in L^2(0,+\infty)\rbrace $ endowed with the norm $||u||^2=\int_0^{+\infty} p(t) u'^2(t) dt$ ...
0
votes
1answer
32 views

Reverse Fatou's lemma on probability space

Let $(\Omega, \mathcal{F},\mathbb{P})$ be probability space and $E_{n \in \mathbb{N}}$ be $\mathcal{F}$-measurable sets. Show example that reverse Fatou's Lemma, $\mathbb{P}(\limsup_n E_n) \geq ...
2
votes
2answers
76 views

Pick a smart function

Our teacher wants us to find a function $f$ on $(0,\pi)$ such that $$\sqrt{\sin(x)} f(x)^{\frac{1}{4}} =k_1 + \cos(x)$$ and $$\sqrt{\sin(x)} f(x)^{-\frac{1}{4}} = k_2 + \cos(x).$$ The two constants ...
3
votes
0answers
48 views
+50

What types of integrals cannot be solved using improper Riemann-Stieltjes Integration?

I came across the wikipedia discussion of the Riemann-Stieltjes integral. The first sentence in the "Generalization" section gave me pause: An important generalization is the Lebesgue–Stieltjes ...
1
vote
0answers
14 views

Change of variables formula with integrator of bounded variation

Let $G$ be be continuous with bounded variation on finite intervals. If $f$ is continuous then it is well known that $\int_a^bf(G(s))dG(s)=\int_{G(a)}^{G(b)}f(x)dx$. How general can $f$ be so that ...
1
vote
1answer
39 views

$B_k$ sequence of closed balls in $\mathbb{R}^n$ such that $B_{k+1}\subset B_k$, show $\cap_k B_k$ is either a point or a closed ball.

$B_k$ sequence of closed balls in $\mathbb{R}^n$ such that $B_{k+1}\subset B_k$, show $\bigcap_k B_k$ is either a point or a closed ball. Please help me check the proof, thanks! Define $x_k$ to be ...
0
votes
1answer
23 views

Understanding Distributional Meanings and Test Functions for PDEs

thank you for taking the time to read my question. My question is about distributional meanings in PDEs. My specific question is at the bottom, but I'd be interested in a bit of general theory (even ...
2
votes
4answers
51 views

Prove a sequence does not tend to zero

Prove that the sequence $\dfrac{a^n}{2^nn^2}$ where $a>2$ does not tend to zero. I thought about writing $a=2+{\epsilon}$ then using binomial expansion which is valid for ${\epsilon}<1$ but ...
0
votes
1answer
25 views

Find the limit $\lim\limits_{n\rightarrow +\infty} n^2 \int_0^{2n} e^{-n|x-n|}\log\left[1+\frac{1}{x+1}\right] dx$

Find the limit $$\lim_{n\rightarrow +\infty} n^2 \int_0^{2n} e^{-n|x-n|}\log\bigg[1+\frac{1}{x+1}\bigg] dx$$ I tried to change the variable $y=n(x-n)$, then we get $$\lim_{n\rightarrow +\infty} n ...
0
votes
1answer
48 views

Prove that $g$ is continuous. [duplicate]

Let $f:[a,b]\to \mathbb{R}$ be a continuous function and let $g:[a,b]\to \mathbb{R}$ be a function such that $g(a)=f(a)$ and $g(x)=\sup_{t\in [a,x]}f(t)$. Prove that $g$ is continuous on $[a,b]$. ...
2
votes
1answer
18 views

Hadamard's Lemma in multidimensional real analysis

This is Hadamard's Lemma: Let $U \subset \Bbb R^n$ be an open set, let $a \in U$ and $f: U \to \Bbb R^p$. Then the following assertions are equivalent. The mapping $f$ is differentiable at $a$. ...
1
vote
1answer
45 views

Subsequences and limit inferior

Suppose $f\colon \mathbb{R} \to \mathbb{R}$ is a continuous function. Let $x\in \mathbb{R}$ and let $(x_n) \subseteq \mathbb{R}$ be a sequence converging to $x$. Let $(y_n)$ be a subsequence of ...
2
votes
1answer
23 views

convolution of compactly supported continuous function with schwartz class function is again a Schwartz class function?

Suppose $f$ is continuous function on $\mathbb R$ with compact support; and $g\in \mathcal{S}(\mathbb R),$ (Schwartz space) My Question is: Can we expect $f\ast g \in \mathcal{S(\mathbb R)}$ ? ...
1
vote
1answer
90 views

Show there exists $x\in (0,1)$ such that $f(x) \leq \int_0^1 f(t) dt$

Please help me check my proof, thanks! (a) Show there exists $x\in (0,1)$ such that $$f(x) \leq \int_0^1 f(t) dt.$$ Proof: when $f$ is constant a.e, the equality holds for all points except for a ...
0
votes
1answer
17 views

$f\in L^{p}(\mathbb R)\cap C_{0}(\mathbb R); (1<p<\infty), g\in C^{\infty}_{c}(\mathbb R) \implies f\ast g \in C^{k}(\mathbb R)$?

We put, $C_{0}(\mathbb R)=$ The space of continuous functions on $\mathbb R$ vanishing at $\infty$; $C^{k}(\mathbb R)=$ The space of all functions $\mathbb R$ whose derivative of order $\leq k$ exist ...
-1
votes
0answers
17 views

Cardinality of the following set of functions on $\mathbb R$ [duplicate]

Consider the following set $W$ = The set of constant functins on $\mathbb R$. $X$ = The set of polynomial functins on $\mathbb R$. $Y$ = The set of continous functins on $\mathbb R$. $Z$ = The set ...
1
vote
1answer
33 views

Proof that this set is not compact

Let $X=C[0,1]$ with the $\sup$ norm. Let $Y = \{f\in X\mid \|f\|_\infty \le 1\}$. It is my goal to show that $Y$ is not compact using the sequence defintion of compactness. Note that it is very easy ...
2
votes
2answers
101 views

How to reduce this to Sturm-Liouville form?

I have the ODE $$-(1-x^2) \frac{d^2 f(x)}{dx^2} + x \frac{df(x)}{dx}+g(x)f(x)=\lambda f(x)$$ and I want to reduce it to Sturm-liouville form. The problem is that we don't have $2x$ but just $x$. ...
4
votes
3answers
109 views

Prove $ne^{-n}$ converges to zero

How would I prove that $ne^{-n}$ converges to zero? I've tried $ne^{-n}<{\epsilon}$ and then logging both sides but no further progress could be made. Thanks
1
vote
1answer
24 views

Equivalence of Localized Fourier Restriction Estimates

I'm reading Tao's Park City notes on the restriction conjecture http://arxiv.org/abs/math/0311181. He says at some point that the estimate: there is a constant $C > 0$ such that, for any $R \ge ...
2
votes
1answer
31 views

Does the limit $\lim_{k\to\infty}\int|\cos kx |f(x) d\lambda(x)$ always exist?

Let $f$ be a Lebesgue integable function. Does the limit $$\lim_{k\to\infty}\int|\cos kx |f(x) d\lambda(x)$$ always exist?
2
votes
1answer
49 views

There exist $c(c\in \left(\frac{1}{2},1\right)$ s.t $2 \int _{0}^{1}f(x)dx=\frac{f(c)}{c} $

I would appreciate if somebody could help me with the following problem: Let $f(0)=0,f(1)=1,f'(x)>0,f''(x)<0$ on $[0,1]$ Q: Under the proposition is true of false? There exist $c(c\in ...
1
vote
0answers
15 views

existence of solution of volterra integral equation of the first kind

$$ \int_0^t k(s,t)f(s)ds=g(t) $$ To know the existence and uniquness of solution of volterra integral equation(VIE) of the first kind, we differentiate it and convert to the VIE of the second kind. ...
-4
votes
0answers
31 views

real anaylsis for twice differentiable function [on hold]

Let f:[0,1] tends to [0,1] be any twice differentiable function satisfying f(ax+(1-a)y) less then or equals to a(f(x)+(1-a)f(y)) for all x,y belongs to [0,1] and any a belongs to [0,1].then for all x ...