Skip to main content
MathBS's user avatar
MathBS's user avatar
MathBS's user avatar
MathBS
  • Member for 6 years, 5 months
  • Last seen more than a month ago
15 votes
2 answers
1k views

How to prove that there exist no functions $f,g:\Bbb{R}\to\Bbb{R}$ such that $f(g(x))=x^{2018}$ and $g(f(x))=x^{2019}$?

8 votes
3 answers
306 views

$f:[0,1]\to[0,1]$ be a continuous function. Let $x_1\in[0,1]$ and define $x_{n+1}={\sum_{i=1}^n f(x_i)\over n}$.Prove, $\{x_n\}$ is convergent

8 votes
3 answers
214 views

$(X,\mu)$ is a measure space. Show that, $L^\infty(X;\mu)$ is either finite dimensional or non-separable.

7 votes
1 answer
227 views

Let $S=\{AB-BA| A,B \in M_n(K)\}$ where $K$ is a field. Prove that $S$ is closed under matrix addition.

6 votes
2 answers
386 views

Condition when angle between two lines is ${\pi\over 3}$

6 votes
2 answers
2k views

Can a vector space over finite field be written as union of finite number of proper subspaces?

6 votes
1 answer
132 views

Let $A$ be a unital $C^*$-algebra, $a\in A,\ x,y\in A_{sa}$. Does there exist a state $\phi$ on $A$ such that $\phi(xa^*ay)=\lVert a\rVert^2\phi(xy)$?

5 votes
3 answers
160 views

Prove that, $\int_{0}^{2\pi}\frac{\cos x+2}{5+4\cos x} dx=\pi$

5 votes
3 answers
262 views

Prove that $\lim_{n \to \infty} \frac{n}{(n!)^\frac{1}{n}} = e $ [duplicate]

5 votes
1 answer
707 views

Let $\alpha>0$. Show that $\sum_{n=1}^\infty {\sin nx\over n^\alpha}$ converges for all $x\in\Bbb{R}$ and examine continuity of the limit function.

4 votes
1 answer
50 views

Let $\phi:C^1([a,b])\to \Bbb{R}$ defined by $\phi(f)=\int\limits_a^b L(t,f(t),f'(t))\ dt$ where $L\in C^1(\Bbb{R}^3)$. Find $D\phi(f)(h)$.

4 votes
1 answer
74 views

Compact subset of banach space is dentable

3 votes
1 answer
68 views

Let $E,F$ be finite dimensional Banach spaces and define $\delta(E,F)=\inf\{\lVert T\rVert\lVert T^{-1}\rVert|\ T:E\to F\text{ is isomorphism}\}$

3 votes
1 answer
79 views

$M_k$ be the set of invertible real $n\times n$ symmetric matrix of index $k$ is open in $\operatorname{Sym}_n(\Bbb{R})$.

3 votes
1 answer
137 views

Let $X$ be a Banach Space.Prove that, $X$ is strictly convex iff every points of $S(X)$ is exposed points of $B(X)$.

3 votes
1 answer
129 views

Prove that $\sum\limits_{p\le x\\ p\text{ prime}} \log p= x+O\left(\frac{x}{\log^2 x}\right)$ using Prime Number Theorem

3 votes
0 answers
57 views

Prove that $\Bbb{Q}(\sqrt2,\sqrt3,u)|_\Bbb{Q}$ is normal where $u^2=(9-5\sqrt3)(2-\sqrt2)$

3 votes
0 answers
56 views

Query regarding the proof of compactness of unit ball of $B(H)$ in weak operator topology where $H$ is a hilbert space.

3 votes
1 answer
69 views

Prove that the map $f:[0,1]\to l_{\infty}([0,1];\Bbb{R})$ defined by $t\mapsto 1_{[t,1]}$ is Riemann integrable.

3 votes
0 answers
2k views

What are the eigenvectors of a $3×3$ real symmetric matrix that admits $(1, 2, 3)^T$ and $(1, 1,−1)^T$ as eigenvectors.

3 votes
1 answer
42 views

Does there exist any $x\in S_{10}$ such that $x(1\ \ 2\ \ 3)x=(9\ \ 10)$?

3 votes
1 answer
115 views

Prove that, there exist uncountably many multiplicative maps from $\Bbb{Q}^\ast\to\Bbb{Z}$

3 votes
3 answers
886 views

Evaluate $\lim_{n\to\infty} [{1\over kn}+{1\over k(n+1)}+{1\over k(n+2)}+\cdots+{1\over k(n+p-k)}]$ where $k<p$

3 votes
1 answer
79 views

$f:[-1,1]\to\Bbb{R}$ be continuous function, and let $g(x)=\int_{0}^{1}f(xy)dy\forall x\in[-1,1]$

3 votes
2 answers
1k views

$f$ is non-vanishing holomorphic function on open unit disc such that $|f(z)|\to1$ as $|z|\to1$. Is $f$ constant?

3 votes
1 answer
140 views

Find a subspace of $L^2([0,1])$ which is dense in $L^p([0,1])$ for all $p<2$ but not in $L^2([0,1])$

3 votes
0 answers
52 views

When the sequence $0\to\Bbb{Z}[G']\to\Bbb{Z}[G]\to\Bbb{Z}[G'']$ is exact given a sequence $G'\to G\to G''$?

2 votes
1 answer
84 views

Let $f\in L^2(\Bbb{T}),\ f\ne0$ and $M=\overline{f\wp_+}$. Prove that, $M$ is simply invariant for $\chi_1$ iff $f\notin \chi_N M$ for some $N>0$

2 votes
0 answers
56 views

Prove that, $H^\infty+C(\Bbb{T})$ is not a $C^*$-algebra

2 votes
1 answer
72 views

$T$ be a finite rank operator of rank $n$. There exist orthonormal set $\{u_1,\ldots,u_n\}$ such that $Th=\sum_{i=1}^n\langle h,u_i\rangle Tu_i$

1
2 3 4 5 6