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

0
votes
1answer
34 views

Analytic Functions and Equicontinuity

Let $r > 0$, $R > 0$, and assume that the power series with real coefficients \begin{equation} \sum_{n,m = 0}^{\infty} a_{n,m} x^{n} y^{m} \end{equation} is absolutely convergent for every real $...
1
vote
1answer
30 views

Determining if this mapping is continuous?

Let $X$ be a closed and bounded subset of $\mathbb{R}^p$ and let $C(X)$ denote the vector space of continuous functions from $X$ to $\mathbb{R}$. For $f,g \in C(X)$, let $$ d_{\infty} (f,g) = \sup \...
1
vote
1answer
37 views

What is the Hilbert adjoint operator of this bounded linear operator?

Let $H$ be a Hilbert space, and let $z \in H$. Let $T_z \colon H \to K$, where $K$ is the field of scalars for $H$ and $K$ is either $\mathbb{R}$ or $\mathbb{C}$, be defined by $$ T_z (x) \colon= \...
1
vote
0answers
25 views

Simplifying Multiple Integral for Compound Probability Density Function

Are there any ways to simplify this multiple integral? $$ \hat{f}\left(\left.y\right|\alpha\right)=\int_{-\infty}^{\infty}\cdots\int_{-\infty}^{\infty}\hat{f}\left(\left.y\right|\theta_{1}\right)\hat{...
10
votes
1answer
87 views

finite polynomials satisfy $|f(x)|\le 2^x$

This is a problem from TsingHua University math competition for high school students. Prove there exists only finite number of polynomials $f\in \mathbb{Z}[x]$ such that for any $x\in \mathbb{N}$ ,...
3
votes
1answer
48 views

Is there a meaning to the notation “\arg \sup”?

When $f$ is a function on a set $A$, the notation: $\arg\max_{x\in A} f(x)$ denotes the set of elements of $A$ for which $f$ attains its maximum value. This set may be empty, for example, if $f(x)=x$ ...
1
vote
0answers
37 views

Is an average function integrable?

I'm thinking about the following question: If $u\in L^p(\mathbb{R}^n)$, is $f(x)=\int_{|y-x|<R}|u(x)-u(y)|^pdy$ in $L^1(\mathbb{R}^n)$, where $R>0$ is a fixed numbers? It's clear that if $u$ ...
3
votes
3answers
36 views

Name for mappings where there is at least one y for every x

There are names for several properties of mappings from $x$ in $X$ to $y$ in $Y$. I think we say that a mapping from X to Y is (a)... Function: there is at most one $y$ for every $x$ Injective: ...
3
votes
1answer
32 views

Alternate proof of the dominated convergence theorem by applying Fatou's lemma to $2g - |f_n - f|$?

Here is a proof of the dominated convergence theorem. Theorem. Suppose that $f_n$ are measurable real-valued functions and $f_n(x) \to f(x)$ for each $x$. Suppose there exists a nonnegative ...
5
votes
1answer
77 views

Open interval $(0,1)$ with the usual topology admits a metric space

which of the following is/are true ? $(0,1)$ with the usual topology admits a metric which is complete . $(0,1)$ with the usual topology admits a metric which is not complete. $...
1
vote
2answers
42 views

Prove $f_n(x)=\frac{x^n}{\sqrt{3n}}$ for $x \in [0,1]$ is uniformly convergent [duplicate]

I'm completely confused by uniform convergence, but I put together the following proof just based on my other questions here and examples I read online. Discussion: Let $\epsilon \gt 0$ We want to ...
7
votes
1answer
37 views

Variant of dominated convergence theorem, does it follow that $\int f_n \to \int f$?

Suppose $f_n$, $g_n$, $f$ and $g$ are integrable, $f_n \to f$ almost everywhere, $g_n \to g$ almost everywhere, $|f_n| \le g_n$ for each $n$, and $\int g_n \to \int g$. Does it follow that $\int f_n \...
4
votes
1answer
27 views

$\sup_n \int |f_n|^{1 + \gamma}d\mu < \infty$ implies $\{f_n\}$ is uniformly integrable?

Suppose $\mu$ is a finite measure and for some $\gamma > 0$, we have$$\sup_n \int |f_n|^{1 + \gamma}d\mu < \infty.$$Does it follow that $\{f_n\}$ is uniformly integrable?
0
votes
1answer
54 views

Diameter of set in metric space

I do agree with the statement that $$d(A) = \sup{\{d(x, y):x, y \in A\}}$$ But why can't we use maximum because according to me its max will also give diameter. I know it should not be correct, so ...
5
votes
3answers
50 views

Show $A\cap B \neq \varnothing \Rightarrow \operatorname{dist}(A,B) = 0$, and $\operatorname{dist}(A, B) = 0 \not\Rightarrow A\cap B \neq \varnothing$

I have a question Let $d$ be a metric on $X$, and define the set to set distance $$\operatorname{dist}(A,B) = \inf\{d(x,y): x\in A, y \in B\}$$ where $A,B \subseteq X$ are nonempty sets ...
2
votes
1answer
55 views

What does it mean in general to show something is well defined? [duplicate]

There is another post that addresses this but quickly fix the problem to be something in arthmetics, and in turn what it means for that arithematics problem to be well defined. I have never ...
2
votes
0answers
44 views

Elementary proof of $C^\infty_c$ is dense in $L^p (L^q)$ mixed space

As it well known, $C^\infty_0 (\mathbb{R}^n)$(the space of infinitely differentiable functions with compact support) is dense in $L^p (\mathbb{R)^n}$ Here I want to consider the same result with ...
-3
votes
1answer
145 views

Again near at Riemann hypothesis [closed]

Let $\zeta(s)$ be Riemann extended zeta function for $Re(s)>0$. Let $\eta(s)$ be Riemann alternated zeta function for $Re(s)>0$, i.e. , $$ \eta(s)=\sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{n^s}=...
7
votes
1answer
53 views

Does there exist a subsequence $n_j$ such that $\int_A f_{n_j}(x)\,dx$ converges for each Borel subset $A$ of $[0, 1]$?

Let $\{f_n\}$ be a sequence of measurable real-valued functions on $[0, 1]$ that is uniformly bounded. Does there exist a subsequence $n_j$ such that $\int_A f_{n_j}(x)\,dx$ converges for each Borel ...
3
votes
1answer
25 views

Does it follow that $\{f_n\}$ is uniformly integrable?

Suppose $\mu$ is a finite measure, $f_n \to f$ almost everywhere, each $f_n$ is integrable, $f$ is integrable, and $\int |f_n - f| \to 0$. Does it follow that $\{f_n\}$ is uniformly integrable?
-1
votes
3answers
54 views

Which is the maximal of an open interval? As [0,2) [closed]

I have the Usual Order defined on $\mathbb{R}$, and I have the interval $[0,2)$. Does it have a maximal element?
1
vote
1answer
63 views

How to prove that : $\int_{2}^{x}\frac{dO(te^{-c\sqrt{\log t}})}{t\log t}=O(1)$ [closed]

$\displaystyle{ \mbox{How I can prove that}\ \int_{2}^{x}{\,\mathrm{d}\alpha\left(t\right) \over t\,\log\left(t\right)} = \,\mathrm{O}\left(1\right)\ ?,}\quad $ where $\alpha\left(t\right) = \,\mathrm{...
6
votes
1answer
71 views

Do we necessarily have that $\int g\,d\mu_n \to \int_0^1 g\,dx$?

Let $\mathcal{B}$ be the Borel $\sigma$-algebra on $[0, 1]$. Suppose $\mu_n$ are finite measures on $([0, 1], \mathcal{B})$ such that $\int f\,d\mu_n \to \int_0^1 f\,dx$ whenever $f$ is a real-valued ...
8
votes
2answers
48 views

$\{f_n\}$ is uniformly integrable if and only if $\sup_n \int |f_n|\,d\mu < \infty$ and $\{f_n\}$ is uniformly absolutely continuous?

Let $(X, \mathcal{A}, \mu)$ be a measure space. A family of measurable functions $\{f_n\}$ is uniformly integrable if given $\epsilon$ there exists $M$ such that$$\int_{\{x : |f_n(x)| > M\}} |f_n(x)...
4
votes
1answer
27 views

Countable collection of Borel subsets of $[0, 1]$, exists subsequence where $\int_A f_{n_j}(x)\,dx$ converges for each $i$?

Let $\{f_n\}$ be a sequence of measurable real-valued functions on $[0, 1]$ that is uniformly bounded. How do I see that if $\{A_i\}$ is a countable collection of Borel subsets of $[0, 1]$, then there ...
5
votes
2answers
94 views

Series of Functions and Continuity

Let $a > 0$, and $(f_n)_{n=0}^{\infty}$ a sequence of continuous functions $f_n:[-a,a] \rightarrow \mathbb{R}$. Assume that the series \begin{equation} \sum_{n=0}^{\infty} x^n f_n(t) \end{equation} ...
1
vote
3answers
36 views

How do I see that if $A$ is a Borel subset of $[0, 1]$, then there exists a subsequence $n_j$ such that $\int_A f_{n_j}(x)\,dx$ converges?

Let $\{f_n\}$ be a sequence of measurable real-valued functions on $[0, 1]$ that is uniformly bounded. How do I see that if $A$ is a Borel subset of $[0, 1]$, then there exists a subsequence $n_j$ ...
2
votes
1answer
21 views

seq. of nonneg. Lebesgue measurable functions on $\mathbb{R}$, have $\limsup_{n \to \infty} \int f_n\,dx \le \int \limsup_{n \to \infty} f_n\,dx$?

Let $f_n$ be a sequence of nonnegative Lebesgue measurable functions on $\mathbb{R}$. Is it necessarily true that$$\limsup_{n \to \infty} \int f_n\,dx \le \int \limsup_{n \to \infty} f_n\,dx?$$If not, ...
2
votes
1answer
48 views

Determine whether $f_n(x)=\frac{x^n}{\sqrt{3n}}$ for $x \in [0,1]$ is uniformly convergent

This is a follow up to my below question: Pointwise convergence: State $f(x) = \lim f_n(x)$ I'm trying to determine whether $f_n(x)=\frac{x^n}{\sqrt{3n}}$ for $x \in [0,1]$ is uniformly convergent. ...
4
votes
1answer
84 views

$\lim_{n \to \infty} \mid a_n + 3(\frac{n-2}{n})^n \mid^{\frac1n} = \frac35$. Then find $\lim_{n \to \infty} a_n$.

Let $\{a_n\}$ be a sequence of real numbers such that $$\lim_{n \to \infty} \mid a_n + 3(\frac{n-2}{n})^n \mid^{\frac1n} = \frac35$$ Then find $\lim_{n \to \infty} a_n$. Tried very hard yet not ...
5
votes
2answers
61 views

$f: \mathbb{R} \to \mathbb{R}$ integrable, $F(x) = \int_a^x f(y)\,dy$, $F$ necessarily continuous

Suppose $f: \mathbb{R} \to \mathbb{R}$ is integrable, and we define$$F(x) = \int_a^x f(y)\,dy.$$Why does it follow that $F$ is necessarily a continuous function?
1
vote
1answer
25 views

Proofs involving limsup and liminf

I've been working with proofs involving $\limsup$ and $\liminf$, and I'm a bit confused regarding their general methodology. More specifically, I'm unsure about whether my approach to the following ...
0
votes
1answer
24 views

Finding a two-dimensional chain

Let $$T=\{(x,y,z,w,)\in R^4:x^2+y^2=z^2+w^2=\frac{1}{\sqrt 2}\}$$ and $$\omega=dx\land dy + dz\land dw$$ in $\mathbb R^4$. How do I find a two-dimensional chain $C$ where $T$ is its trace? And how can ...
3
votes
1answer
57 views

Find the limit of $\lim\limits_{(x,y)\to (0,0)} \frac{x^2y^2}{x^2y^2+(x-y)^2}$

Find the limit of $$\lim\limits_{(x,y)\to (0,0)} \frac{x^2y^2}{x^2y^2+(x-y)^2}$$ So, I know that $$\lim\limits_{x \to x_0} f(x)=c \Leftrightarrow \forall (x_n)\subseteq D\setminus\{x_0\}, x_n\to x_0:...
0
votes
1answer
26 views

Convergence radius for power series $ y(x)=\sum^{\infty}_0 a_nx^n\;$ with recurrence relation [on hold]

Suppose the power series $ y=\sum^{\infty}_0 a_nx^n\;$ has terms that satisfies the recurrence relation: $$a_{n+2}=\frac{(2n_1)(3n+2)}{(n+3)(2n-5)}a_n$$ with $a_0=1$,$a_1=0$. What is the radius of ...
1
vote
1answer
27 views

Puzzled over terminology “Indefinite Integral” in Royden

In Royden (pg 125), a function of the form $$f(x)=f(a)+\int_a^x f'$$ is called an "indefinite integral" of $f'$ over $[a,b]$. However, I don't see how it is "indefinite", I thought indefinite means ...
1
vote
1answer
28 views

Pointwise convergence: State $f(x) = \lim f_n(x)$

I'm completely confused by this subject and hoping you guys can help me to clear up my confusion. So I'm told: State $f(x) = \lim f_n(x)$ where $f_n(x)=\frac{x^n}{\sqrt{3n}}$ for $x \in [0,1]$ ...
2
votes
1answer
58 views

Lower bound for derivative of a non-constant function

Here is the problem. Let $g$ be a non-constant differentiable real function on $[a,b]$, and we know $g(a) = g(b) = 0$. I want to prove that there is a $c\in(a,b)$ such that $$\lvert g'(c)\rvert > \...
3
votes
1answer
39 views

Mean Value Theorem used to prove “global properties”?

This is a soft question: I am puzzled by this statement on Wikipedia: https://en.wikipedia.org/wiki/Mean_value_theorem: The theorem is used to prove global statements about a function on an ...
0
votes
2answers
31 views

Zeros and poles of some meromorphic 1-forms on the riemann sphere

Let $X=\mathbb C_{\infty}$ be the Riemann sphere with the local coordinates $\{z\ ,1/z\}$. I want to show the following two statements: i) There does not exist any non-vanishing holomorphic 1-form on ...
-2
votes
1answer
38 views

Integrating differential forms on curves [closed]

How can I integrate the differential form $$\omega=x\,dx+y\,dy+z\,dz$$ in $\mathbb R^3$ on the curve $$c:[0,2\pi]\to\mathbb R^3: t\mapsto (e^{t\sin t}, t^2-2\pi t, \cos \frac{t}{2})?$$ Some advice ...
1
vote
0answers
23 views

A non-constant holomorphic map $F$ between riemann-surfaces is an isomorphism

I want to show the following: Let $F:X\rightarrow Y$ be a non-constant and holomorphic map between compact riemann surfaces with $genus(X)=genus(Y)\geq 2$. In the above it holds that $F$ is an ...
14
votes
2answers
761 views

Bounding $\int_0^1 f(x) dx $ under the condition $\int_0^1 f'(x)^2 dx \le 1$

Any tips on how to solve this? Problem 1.1.28 (Fa87) Let $S$ be the set of all real $C^1$ functions $f$ on $[0, 1]$ such that $f(0) = 0$ and $$\int_0^1 f'(x)^2 dx \le 1 \;. $$ Define ...
2
votes
0answers
19 views

Upper estimate of function such that $\int |x|^2 f(x) dx<\infty$

Let $f$ be a non-negative function on $\mathbb R^d$ satisfying the following: (1) There exists a non-increasing function $g$ on $(0,\infty)$ such that (1-i) $ C_1^{-1} g(|x|) \le f(x) \le C_1 g(|x|)...
0
votes
1answer
26 views

How do you prove triangle inequality for this metric?

Let $f: \mathbb{R}^+ \rightarrow \mathbb{R}^+$ be an increasing concave function such that $f(t) = 0$ if and only if $t = 0$. Let $(X,d)$ be a metric space. Show that $d_f = f \circ d $ defines a ...
0
votes
1answer
19 views

Helly theorem application

Let $X$ be a normed space, $dim(X)=d$, let $r>0, A$ and $\subset X$ . Show that if every $d+1$ points of $A$ are contained in a closed ball of radius $r$, then $A$ is contained in a closed ball of ...
7
votes
1answer
66 views
+50

Intuition behind proof of bounded convergence theorem in Stein-Shakarchi

Theorem 1.4 (Bounded convergence theorem) Suppose that $\{f_n\}$ is a sequence of measurable functions that are all bounded by $M$, are supported on a set $E$ of finite measure, and $f_n(x) \to f(x)$ ...
2
votes
0answers
20 views

How to identify properties of the zeroes of this polynomial? [on hold]

If $f_0(x)=1$, and $f_{n+1}=\frac{d}{dx}((x^2-1)f_n(x))$, prove that every $f_n$ has exactly $n$ distinct zeroes, all located in the interval $(-1,1)$. It's got me stumped, so any help/pointers would ...
4
votes
1answer
42 views

$f_n \to f$ almost everywhere and $\int |f_n| \to \int |f|$ implies $\int |f_n - f| \to 0$?

Suppose $f_n$ and $f$ are integrable, $f_n \to f$ almost everywhere, and $\int |f_n| \to \int |f|$. Does it necessarily follow that$$\int |f_n - f| \to 0?$$
2
votes
1answer
25 views

Integral of sequence converges? [closed]

Suppose $(X, \mathcal A, \mu)$ is a measure space, $f$ and each $f_n$ is integrable and nonnegative, $f_n \to f$ almost everywhere, and $\int_X f_n \to \int _X f$. Does it necessarily follow that for ...