User illysial

28 votes

4 answers

28k views

The closure of a connected set in a topological space is connected

general-topology connectedness

asked Feb 16, 2014 at 2:57

13 votes

1 answer

484 views

Will the Lebesgue integral of a real valued function always be a Riemann sum?

real-analysis measure-theory

asked Sep 4, 2015 at 3:08

9 votes

3 answers

853 views

Prove that $\lim_{t\to \infty} t\mu(\{x:f(x)\geq t\})=0$

real-analysis integration measure-theory

asked Jun 24, 2015 at 5:55

8 votes

1 answer

2k views

If $\mu$ is finite, then $\{f_n\}$ is uniformly integrable iff $\sup_n \int|f_n| d\mu<\infty$ and $\{f_n\}$ is uniformly absolutely continuous.

measure-theory proof-verification lebesgue-integral

asked Jun 19, 2015 at 20:27

6 votes

1 answer

312 views

A collection $\{f_\alpha\}_{\alpha \in A}$ so that $\sup_{\alpha \in A} f_{\alpha}(x)$ is finite and non-measurable

measure-theory lebesgue-measure

asked Jun 13, 2015 at 19:01

5 votes

1 answer

2k views

Composition of 2 Lebesgue measurable functions is not lebesgue measurable: Are these two functions Borel Measurable?

measure-theory

asked Jun 14, 2015 at 1:03

5 votes

3 answers

541 views

What is wrong with my application of Lebesgue Dominated Convergence Theorem in these two examples?

measure-theory lebesgue-integral

asked Jun 17, 2015 at 23:28

5 votes

1 answer

202 views

Question about the definition of the dual map of the differential

differential-geometry

asked Mar 18, 2016 at 20:35

4 votes

0 answers

332 views

Understanding the construction of Exterior Algebra

linear-algebra abstract-algebra differential-geometry tensor-products multilinear-algebra

asked Apr 3, 2016 at 1:47

4 votes

1 answer

972 views

Suppose $\mu$ is a finite measure and $\sup_n \int |f_n|^{1+\epsilon} \ d\mu<\infty$ for some $\epsilon$. Prove that $\{f_n\}$ is uniformly integrable

measure-theory proof-verification lebesgue-integral

asked Jun 21, 2015 at 5:06

4 votes

1 answer

317 views

Let $f_1,f_2,\ldots, f_n$ be linear functionals on $X$. Show $f=\sum_{i=1}^n\lambda_i f_i$ iff $\bigcap \ker f_i \subset \ker f$

functional-analysis

asked Jul 18, 2015 at 23:38

4 votes

1 answer

174 views

Find the exact value of $t$ that maximizes $\int_{t}^{t+1}\sin e^x \ dx$

calculus

asked May 5, 2014 at 22:46

4 votes

2 answers

106 views

Sufficient Condition for $\lim_{n\rightarrow \infty} \frac{a_n}{b_n}=1$

real-analysis

asked Aug 5, 2014 at 0:53

3 votes

1 answer

578 views

Solve $3^x+3^{(3x+1)}=108$

algebra-precalculus

asked Aug 8, 2014 at 0:12

3 votes

4 answers

2k views

Fixed point for a continuous function on a compact set?

real-analysis compactness fixed-point-theorems

asked Aug 9, 2014 at 3:06

3 votes

3 answers

227 views

A function $F:\mathbb{Q} \rightarrow \rm numerators$

number-theory

asked Apr 3, 2014 at 2:11

3 votes

1 answer

344 views

Counting Periodic Orbits on a regular Hexagon

combinatorics geometry billiards

asked Apr 5, 2014 at 7:08

3 votes

1 answer

3k views

L'hopital's Rule in higher dimensions.

multivariable-calculus

asked Jul 13, 2014 at 0:53

3 votes

0 answers

74 views

Given a point $A$, describe those points to which a catenary cannot be drawn from $A$.

calculus calculus-of-variations

asked Sep 13, 2014 at 23:59

3 votes

1 answer

434 views

Concerning existence of subsequence of converging integrals on subsets of $[0,1]$ of a sequence $(f_n)\in[0,1]$

measure-theory lebesgue-integral

asked Jun 18, 2015 at 20:15

3 votes

1 answer

1k views

Show that $\Gamma(z)\Gamma(1-z) \sin \pi z$ is bounded in the complex plane

complex-analysis gamma-function beta-function

asked Mar 1, 2016 at 4:23

3 votes

2 answers

493 views

Using Universal Mapping Property of Tensors to show that $\dim V \otimes W=(\dim V)(\dim W)$

linear-algebra abstract-algebra tensor-products

asked Apr 2, 2016 at 3:12

3 votes

1 answer

64 views

Prove $\int_0^{t_1}\int_0^{\tau}f(s)\ ds\ d\tau=\int_0^{t_1}(t_1-s)f(s)ds$

calculus integration

asked Sep 9, 2015 at 1:02

2 votes

1 answer

51 views

If $\mathrm{min}_{u\in U}||v-u||=||v-u_0||$, then $v-u_0\in U^{\perp}$

linear-algebra inner-products

asked Sep 5, 2015 at 23:46

2 votes

0 answers

197 views

$f_n \rightarrow 0$ weakly if and only if $(\|f_n\|)_{n=1}^{\infty}$ is bounded and $f_n$ converges pointwise to $0$.

functional-analysis measure-theory lebesgue-integral

asked Jul 10, 2015 at 0:59

2 votes

1 answer

49 views

How is the Chain rule used to show that $\int_t^{t+\tau} f(x(t))\frac{dx(t)}{dt} \ dt=\int_{x(t)}^{x(t+\tau)} f(x) \ dx$?

calculus

asked Nov 19, 2014 at 19:30

2 votes

2 answers

75 views

Prove that if $f$ is linear, then for any $\textbf{a},\textbf{v} \in \mathbb{R}^2$, $f(\textbf{a}+\textbf{v})=f(\textbf{a})+[Df(\textbf{a})]\vec{v}$

linear-algebra multivariable-calculus

asked Sep 11, 2014 at 19:18

2 votes

1 answer

114 views

Show that $g(x)=\text{mid}\{f_1,f_1,f_3\}$ is measurable.

real-analysis

asked Apr 14, 2015 at 16:26

2 votes

1 answer

135 views

If $X=\{0,1\}$, there exists an outer measure $\mu^$ on $X$ such that $\mu^ \neq \mu^+$

real-analysis measure-theory

asked May 24, 2015 at 4:02

2 votes

1 answer

389 views

If $f=g$ on $(x-r,x+r)\cap (0,1)$ and $g$ is Borel measurable, then $f$ is Borel measurable.

measure-theory

asked Jun 11, 2015 at 18:04

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52 Questions

The closure of a connected set in a topological space is connected

Will the Lebesgue integral of a real valued function always be a Riemann sum?

Prove that $\lim_{t\to \infty} t\mu(\{x:f(x)\geq t\})=0$

If $\mu$ is finite, then $\{f_n\}$ is uniformly integrable iff $\sup_n \int|f_n| d\mu<\infty$ and $\{f_n\}$ is uniformly absolutely continuous.

A collection $\{f_\alpha\}_{\alpha \in A}$ so that $\sup_{\alpha \in A} f_{\alpha}(x)$ is finite and non-measurable

Composition of 2 Lebesgue measurable functions is not lebesgue measurable: Are these two functions Borel Measurable?

What is wrong with my application of Lebesgue Dominated Convergence Theorem in these two examples?

Question about the definition of the dual map of the differential

Understanding the construction of Exterior Algebra

Suppose $\mu$ is a finite measure and $\sup_n \int |f_n|^{1+\epsilon} \ d\mu<\infty$ for some $\epsilon$. Prove that $\{f_n\}$ is uniformly integrable

Let $f_1,f_2,\ldots, f_n$ be linear functionals on $X$. Show $f=\sum_{i=1}^n\lambda_i f_i$ iff $\bigcap \ker f_i \subset \ker f$

Find the exact value of $t$ that maximizes $\int_{t}^{t+1}\sin e^x \ dx$

Sufficient Condition for $\lim_{n\rightarrow \infty} \frac{a_n}{b_n}=1$

Solve $3^x+3^{(3x+1)}=108$

Fixed point for a continuous function on a compact set?

A function $F:\mathbb{Q} \rightarrow \rm numerators$

Counting Periodic Orbits on a regular Hexagon

L'hopital's Rule in higher dimensions.

Given a point $A$, describe those points to which a catenary cannot be drawn from $A$.

Concerning existence of subsequence of converging integrals on subsets of $[0,1]$ of a sequence $(f_n)\in[0,1]$

Show that $\Gamma(z)\Gamma(1-z) \sin \pi z$ is bounded in the complex plane

Using Universal Mapping Property of Tensors to show that $\dim V \otimes W=(\dim V)(\dim W)$

Prove $\int_0^{t_1}\int_0^{\tau}f(s)\ ds\ d\tau=\int_0^{t_1}(t_1-s)f(s)ds$

If $\mathrm{min}_{u\in U}||v-u||=||v-u_0||$, then $v-u_0\in U^{\perp}$

$f_n \rightarrow 0$ weakly if and only if $(\|f_n\|)_{n=1}^{\infty}$ is bounded and $f_n$ converges pointwise to $0$.

How is the Chain rule used to show that $\int_t^{t+\tau} f(x(t))\frac{dx(t)}{dt} \ dt=\int_{x(t)}^{x(t+\tau)} f(x) \ dx$?

Prove that if $f$ is linear, then for any $\textbf{a},\textbf{v} \in \mathbb{R}^2$, $f(\textbf{a}+\textbf{v})=f(\textbf{a})+[Df(\textbf{a})]\vec{v}$

Show that $g(x)=\text{mid}\{f_1,f_1,f_3\}$ is measurable.

If $X=\{0,1\}$, there exists an outer measure $\mu^$ on $X$ such that $\mu^ \neq \mu^+$

If $f=g$ on $(x-r,x+r)\cap (0,1)$ and $g$ is Borel measurable, then $f$ is Borel measurable.