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.

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2
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4answers
55 views

$f\in L^2([0,1])$ and $\int_0^1 t^n f(t)\,dt = (n+2)^{-1}$ for all $n=0,1,2,\ldots$. Is $f(t)=t$ a.e.?

I know how to prove the statement if $f$ is continuous, but the $L^2$ part is throwing me off. As far as I know, we can't use a version of Stone-Weierstrass, because $f$ isn't continuous. I'm pretty ...
1
vote
1answer
30 views

When is the following inequality true? $ \int_0^1 \lvert f-g\rvert \leqslant \max_{x\in[0,1]}\lvert f(x)-g(x)\rvert $

Let $ f,g \in \mathcal{C}([0,1])$. Then we have $$ \int_0^1 \lvert f(x)-g(x)\rvert dx \leqslant (1-0)\max_{x\in[0,1]}\lvert f(x)-g(x)\rvert $$ Now, is it only true for continuous functions in $[0,...
2
votes
1answer
48 views

Modified Wave Equation: Bound $\int u^2 \, dx$

I'm studying for a qualifying exam and I can't figure this problem out: Suppose $B \subset \mathbb R^n$ is the unit ball centered at the origin and that $u$ is a smooth solution of \begin{align*} ...
0
votes
1answer
34 views

line integral of 3D vector field

Suppose I have a 3D vector field $\vec v(x,y,z)=(v_1,v_2,v_3)$ and I want to compute $$\int_C \vec{v}\cdot \vec n\, dS$$ where $C$ is the unit circle $C\equiv\{(x,y)\in\mathbb{R}^2\,:\, x^2+y^2=1\}$ ...
2
votes
2answers
34 views

$W^{1,1}\subseteq AC$ and a certain property implies BV: why?

Brézis states that the functions in $W^{1,1}(I)$, with $I$ a bounded interval, are absolutely continuous, and that, for $u\in L^1(I)$, if the following holds for some constant $C$: $$\left|\int_Iu\...
1
vote
1answer
21 views

In a metric space $X$, a subset $S \subset X$ is relatively sequentially compact if and only if its closure $\overline{S}$ is sequentially compact.

Prove that in a metric space $X$, a subset $S \subset X$ is relatively sequentially compact if and only if its closure $\overline{S}$ is sequentially compact. The terms relatively sequentially ...
1
vote
0answers
25 views

A problem about a continuous iterated function [duplicate]

Let $f:\mathbb {R} \rightarrow \mathbb { R } $ be a continuous function such that $f\circ f \circ f=\text{id}_\mathbb{R} $. Show that $f=\text{id}_\mathbb{R}$. Is there any hint to prove this? ...
2
votes
3answers
48 views

is it possible for a two divergent series have a convergant diffrence?

I am working on a series where I have split it into 2. the first is the sum of a sequence $a_n$ that converge to $1/2$. and the second $b_n$ is the same thing. is it possible for their difference $\...
0
votes
1answer
14 views

A question on convergence of sets

Thinking back to some fond memories of real and linear analysis last fall, I recalled that a sequence of sets $\{A_{k}\}_{k \in \mathbb{N}}$ where $A_{k} \subseteq \mathbb{R}^n$ for all $k$ converges ...
5
votes
2answers
74 views

Real Analysis, Folland Theorem 3.18 Differentiation on Euclidean Space

Background Information: A measurable function $f:\mathbb{R}^n\rightarrow \mathbb{C}$ is called locally integrable (w.r.t Lebesgue measure) if $\int_K |f(x)|dx < \infty$ for every bounded ...
8
votes
4answers
364 views

Does the series $1-\frac12+\frac12-\frac1{2^2}+\frac13-\frac1{2^3}+\frac14-\frac1{2^4}+\frac15-\frac1{2^5}+\cdots$ converge or diverge?

$1-\frac{1}{2}+\frac{1}{2}-\frac{1}{2^2}+\frac{1}{3}-\frac{1}{2^3}+\frac{1}{4}-\frac{1}{2^4}+\frac{1}{5}-\frac{1}{2^5}+\cdots$ I've be trying to figure out how to write this series symbolically so I ...
3
votes
3answers
29 views

Why does $A_1^\text{c}$ have an infinite number of measurable subsets?

Let $\mathcal{A}$ be a $\sigma$-algebra. Show that if $|\mathcal{A}| = \infty$, then $\mathcal{A}$ is uncountable. We want to construct an infinite sequence of nonempty disjoint measurable sets. ...
1
vote
1answer
13 views

Inverse of a vector of functions

I know how can I find the inverse of a function , but I am really confused when I deal with a vector of functions Suppose I have this vector : $$ g(x_1,x_2)= \begin{bmatrix} x_1 +x_2\\x_2-x_1\end{...
1
vote
2answers
45 views

Which of the following subsets of $\Bbb R^2$ are homeomorphic to the set $\{(x, y) \in \Bbb R^2 \mid xy = 1\}$?

Which of the following subsets of $\Bbb R^2$ are homeomorphic to the set $\{(x, y) \in \Bbb R^2 \mid xy = 1\}$? $a. \{(x, y) ∈ \Bbb R^2 \mid xy − 2x − y + 2 = 0\}.$ $b. \{(x, y) ∈ \Bbb R^2 \...
5
votes
2answers
37 views

Assuming $\sum_{n = 1}^\infty \int |f_n| < \infty$, properties that follow for integral

How do I see that if $\sum_{n = 1}^\infty \int |f_n| < \infty$, then $\sum_{n = 1}^\infty f(x)$ converges absolutely almost everywhere, is integrable, and its integral is equal to $\sum_{n = 1}^\...
2
votes
3answers
32 views

Show the countability of the non-zero elements when sum of elements is less than $\infty$ on the extended reals.

Show that for $(x_{\alpha})_{\alpha \in A}, x_{\alpha} \in [0, +\infty]$ where $\sum_{\alpha \in A}x_{\alpha} < \infty$ the number of non-zero elements is at most countable. Note $A$ can be ...
1
vote
0answers
55 views

Norm of gradient of velocity field

If $\mathbf{u}(x,y,z,t)=(u,v,w):\mathbb{R}^3\times[0,+\infty)\to\mathbb{R}^3$ denotes a velocity field, what is the definition for $\|\nabla\mathbf{u}\|_{L^{\infty}}$? I know that $\nabla\mathbf{u}$ ...
0
votes
0answers
16 views

Is the Fourier transform a unitary isomorphism between $L^2(\mathbb{T}^n)$ and $\ell^2(\mathbb{T}^n)$

I am reading through Folland's "Real Analysis", and it's clear that if $f\in L^2(\mathbb{T}^n)$, then $\{\hat{f}(\kappa)\}\in\ell^2(\mathbb{T}^n)$, and the norms of those two are equal. However, it's ...
6
votes
3answers
69 views

Show that linear functional $L(f) = \int_0^1 f(x) dx$ is continuous

Let $(C[0,1], d_1)$ be a metric space of all continuous functions $f:[0,1] \to \mathbb{R}$, $d_1$ is the $L_1$ metric $$d_1(f,g) = \int\limits_0^1 |f(x) - g(x)| dx$$ Show that linear functional $L(...
0
votes
1answer
56 views

Integrate function by partial derivative

I'm searching a $\phi(x,t)$ solution of a pde cauchy system, with $x\in[-1,1],t\in[0,T]$ I am able to know: a) $\phi(x,0)=-cos\left(\pi\left(x-0.85\right)\right)$ b) $\phi_x(x,t)$, $\forall t,x$ (...
-1
votes
0answers
34 views

Exponential equations in one variable for the reals [closed]

My father approached me yesterday and asked me if I could solve $$4^{x}+5^{x}=6^{x}$$ I countered by asking him over what set. He told me $R_{>0}$ So by using the intermediate value theorem it's ...
1
vote
0answers
33 views

Symbol of differential operator and change of coordinates

Some time ago I posted the question about the change of coordinates in differential operator. Here is the relevant discussion Symbol of differential operator transforms like a cotangent vector The ...
1
vote
1answer
48 views

Show all sequence of $l^1$ with $|x_n|\leq \frac{1}{n^2}$ is compact.

Could you help me to check my proof: let $\{x^k\}$ be a sequence in such set, we use Cantor's diagonal argument to show the existence of convergent subsequence. There exists a subsequence $\{x^{\...
1
vote
1answer
50 views

Differentiability of a C^2 function

The problem is as follows: Let $U$ be an open set of $\mathbb{R}^m$ and consider $f : U \rightarrow \mathbb{R}$ a function of class $C^2$. Show that for every $x \in U$, $v, w \in \mathbb{R}^m$, $$ f'...
0
votes
0answers
30 views

Simplify an Integration with cosine, logarithm and hyperbolic functions

I would like to simplify (or solve) the following integral: $$F(t):=\int_{a}^{t-a} \cos\left(\ln\left( \frac{f(t-x)}{f(x)} \right)\lambda \right)\frac{1}{\sqrt{f(t-x)\cdot f(x)}} dx, \quad t>2a,$$ ...
0
votes
0answers
10 views

Questions about the regularity of the solution of the heat equation in a bounded domain

I have questions about the proof of the following theorem: Theorem 8 (Smoothness). Suppose $u\in C^2_1(U_T)$ solves the heat equation in $U_T$. Then $u\in C^\infty(U_T)$ Here is the statement and ...
1
vote
0answers
44 views
+50

Following conditions for convergence of measures equivalent

Let $\mathcal{B}$ be the Borel $\sigma$-algebra on $[0, 1]$. Let $\mu_n$ be a sequence of finite measures on $([0, 1], \mathcal{B})$ and let $\mu$ be another finite measure on $([0, 1], \mathcal{B})$. ...
1
vote
1answer
60 views

Does $\sum \frac{1}{n^x}$ converge uniformly?

$\sum \frac{1}{n^x}$ for $x \in \left[\frac{4}{\pi},\infty\right)$ converge uniformly. I'm trying to use the Weierstrass M-Test but having trouble finding an $M_n$. Any hints?
2
votes
1answer
36 views

Does it follow that two finite positive measures are the same?

Suppose $\mu$ and $\nu$ are finite positive measures on the Borel $\sigma$-algebra on $[0, 1]$ such that $\int f\,d\mu = \int f\,d\nu$ whenever $f$ is real-valued and continuous on $[0, 1]$. Does it ...
1
vote
0answers
54 views

Let $u_n > 0 ,v_n > 0$ for all $n$ which are bounded, then $(\lim \sup u_n)\cdot(\lim \sup v_n) \geq \lim \sup u_n v_n$.

The question is : Let $\{u_n\}$ and $\{v_n\}$ be two bounded sequences such that $u_n > 0$ and $v_n > 0$ for all $n \in \mathbb {N}$, then show that $(lim \sup u_n).(lim \sup v_n) \geq lim \...
1
vote
0answers
34 views

Show $N_p[f]=(\frac{1}{|E|}\int_{E}|f|^p)^{\frac{1}{p}}$ is monotone in $p$

For $0<p\leq \infty$ and $0<|E|<\infty$ ($|E|$ is the lebesgue measure of $E$), define $$ N_p[f]= \left( \frac{1}{|E|} \int_E |f|^p \right)^{1/p}, $$ where $N_\infty[f]$ means $\|f\|_\infty=...
3
votes
1answer
32 views

Does it follow that $\mu$ is a measure? [duplicate]

Suppose $\mu_n$ is a sequence of measures on $(X, \mathcal{A})$ such that $\mu_n(X) = 1$ for all $n$ and $\mu_n(A)$ converges as $n \to \infty$ for each $A \in \mathcal{A}$. Cal the limit $\mu(A)$. ...
3
votes
1answer
51 views

Contiuous function on a closed bounded interval is uniformly continuous. Don't understand the proof.

I'm self studying real analysis from Wade's "An Introduction to Real Analysis" and I've come across a proof that I don't understand. I was hoping that some might be able to walk me through it. The ...
2
votes
2answers
52 views

Cantor's theorem about countable sets

Let $f:\mathbb{N} \rightarrow \mathbb{R}$ be a sequence of real numbers. Cantor's theorem states that no interval (a,b) can be the range of a sequence of real numbers. I know the proof for this ...
26
votes
2answers
361 views

If $f$ is a smooth real valued function on real line such that $f'(0)=1$ and $|f^{(n)} (x)|$ is uniformly bounded by $1$ , then $f(x)=\sin x$?

Let $f : \mathbb R \to \mathbb R$ be a smooth ( infinitely differentiable everywhere ) function such that $f '(0)=1$ and $|f^{(n)} (x)| \le 1 , \forall x \in \mathbb R , \forall n \ge 0$ ( as usual ...
11
votes
2answers
466 views

Differentiating the binomial coefficient

I took a lecture in combinatorics this semester and the professor did the following step in a proof: He showed that function $f: x \mapsto \binom{x}{r}$ is convex for $x > r - 1$ (in order to use ...
3
votes
1answer
48 views

Power Series in Two Variables and Radius of Convergence

Let $\alpha > 0$, $\beta > 0$, 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 ...
0
votes
1answer
26 views

The limit of weighted partial sums

Let $f_n(k)$ be a function defined on $k\in\{0,1,2,\dots,n\}$, such that $\lim_{n\rightarrow\infty}f_n(k)=f(k)$ uniformly and $\lim_{k\rightarrow\infty}f(k)=0$. Also, let $w_n(k)$ be a weight ...
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
49 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
37 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
78 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 \...