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Janitha357
  • Member for 7 years, 6 months
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13 votes
2 answers
2k views

How to identify limit ordinals?

11 votes
3 answers
6k views

Prove that $(l^\infty,\|.\|_\infty)$ is a Banach space.

11 votes
2 answers
2k views

Show that if $E$ is not measurable, then there is an open set $O$ containing $E$ that has finite outer measure and for which $m^*(O-E)>m^*(O)-m^*(E)$

9 votes
1 answer
737 views

Does the Dorroh Extension Theorem simplify ring theory to the study of rings with identity?

8 votes
1 answer
598 views

Prove that $\lim_{n\to\infty}\int f_n=\int f$.

7 votes
1 answer
193 views

Calculate $\lim_{n\to\infty}\int_{[0,n]}(1+\frac{x}{n})^ne^{-2x}dx$ .

6 votes
1 answer
9k views

Prove that any interval is measurable.

5 votes
1 answer
141 views

Prove that $I=I_1\oplus I_2\oplus\cdots\oplus I_n$.

5 votes
2 answers
3k views

Prove that for any $\epsilon >0$ there exists a measurable set $E$ such that $m(E)<\infty$ and $\int_E f>(\int f)-\epsilon$.

4 votes
1 answer
74 views

Assume that $\forall a\in R, a^2+a\in \operatorname{cent}R$. Need to show that $R$ is a commutative ring.

4 votes
1 answer
1k views

$\sum_{n=1}^\infty\frac{z^n}{n}$ does not converge uniformly on $\mathbb{D}$.

4 votes
2 answers
402 views

Prove that either $F\subseteq \operatorname{ker} f$ or else $R^\prime$ contains a subring isomorphic to $F$.

4 votes
1 answer
645 views

Prove that $D:C^1[0,1]\to C[0,1]$ is not continuous.

3 votes
1 answer
5k views

Prove that $(l^p,\|.\|_p)$ is a Banach space for $p\geq 1$.

3 votes
0 answers
185 views

Prove that $(S,\|.\|_\infty)$ is not complete.

3 votes
0 answers
1k views

Prove that every ideal of $M_n(R)$ is of the form $M_n(I)$, where $I$ is an ideal of $R$.

3 votes
1 answer
756 views

Show that $A$ is an ideal of $R$.

3 votes
3 answers
4k views

Prove that if $(f_n)$ converges to $f$ in measure then $(f_n^2)$ converges to $f^2$ in measure.

3 votes
0 answers
98 views

If $\sum\limits_nz_n$ converges and $|\arg(z_n)|\leq\theta<\frac{\pi}{2}$ for every $n$ then $\sum\limits_nz_n$ converges absolutely [duplicate]

3 votes
1 answer
2k views

Show that if $p$ is a prime satisfying $n<p<2n$ then $\binom{2n}{n}\equiv 0 \pmod p$.

3 votes
0 answers
158 views

Show that $\lim_{n\to\infty}\int_{[0,1]}\frac{nx}{1+n^2x^2}=0$.

2 votes
0 answers
103 views

Prove that $f$ can be continuously extended over $X$.

2 votes
1 answer
299 views

Calculate $\lim_{n\to\infty}\int_{[0,1]}\frac{nx}{1+nx^2}$.

2 votes
1 answer
162 views

Prove that there exists a continuous function $f:\mathbb{R}\to\mathbb{R^2}$ such that $f(\mathbb{Z})=\mathbb{Z^2}$

2 votes
0 answers
41 views

How to show $charR=lcm\{a_1,...,a_m\}$?

2 votes
1 answer
94 views

A question related to product topology

2 votes
0 answers
266 views

If $f$ is a real valued function on $\mathbb{R}$ such that $\{x|f(x)=a\}$ is measurable for each $a\in\mathbb{R}$ then $f$ is measurable?

2 votes
0 answers
58 views

$\{x\in E|f(x)<r\}$ is measurable for each $r\in\mathbb{Q}$. Is $f$ necessarily measurable?

2 votes
1 answer
78 views

How to retain the key points of an exercise?

2 votes
0 answers
496 views

Show that each ring with identity and characteristic zero contains a subring isomorphic to $\mathbb{Z}$ [duplicate]