Reputation
38,469
Next tag badge:
961/1000 score
409/200 answers
Badges
5 41 113
Newest
 Necromancer
Impact
~319k people reached

Jul
31
comment Infinite product of sine function
@PhoemueX this was a sloppy definition, now fixed
Jul
31
revised Infinite product of sine function
added 23 characters in body
Jul
31
reviewed Approve Infinite product of sine function
Jul
30
comment Evaluate $\displaystyle\lim_{n \to \infty} \int_{0}^1 [x^n + (1-x)^n ]^\frac{1}{n} \ \mathrm{d}x$
Oh, I thought $\{\}$ parenthesis denoted fractional part here. Otherwise ignore first hint and immediately apple the second and the third ones.
Jul
30
comment Evaluate $\displaystyle\lim_{n \to \infty} \int_{0}^1 [x^n + (1-x)^n ]^\frac{1}{n} \ \mathrm{d}x$
1 Fractional part is redundant here. 2 Apply dominated convergence theorem 3 Note that $\lim_{n\to\infty}(a^n+b^n)^{1/n}=\max(a,b)$ for positive $a$ and $b$.
Jul
30
revised $\ell^p$ as a direct summand of $L^p$
added 102 characters in body
Jul
30
answered $\ell^p$ as a direct summand of $L^p$
Jul
25
comment The space of continuous functions as a dual space
@AlexM.exactly.
Jul
25
comment The space of continuous functions as a dual space
@AlexM. As for your question in comments it is completely different. In fact the space of continuous functionals on $X^*$ endowed with weak* topology is exactly $X$. It is remains to note that for $X=C(K)$ we have $X^*=M(K)$.
Jul
25
comment The space of continuous functions as a dual space
Do you mean isometric isomorphism or arbitrary isomorphism of Banach spaces?
Jul
24
comment How to show that quotient space $X/Y$ is complete when $X$ is Banach space, and $Y$ is a closed subspace of $X$?
Because you can put $n=n_k$ in the inequality $\Vert x_n-x_{n_k}\Vert<2^{-k}$.
Jul
24
comment How to show that quotient space $X/Y$ is complete when $X$ is Banach space, and $Y$ is a closed subspace of $X$?
@Lukkio. Ok let's try it other way. There exists $n_1$ such that $\Vert x_n-x_{n_1}\Vert<2^{-1}$ for all $n>n_1$. By induction we can show that, for each $k>1$ we can find $n_k>n_{k-1}$ such that $\Vert x_n-x_{n_k}\Vert<2^{-k}$ for all $n\geq n_k$. These $n_k$'s are the desired ones.
Jul
24
comment How to show that quotient space $X/Y$ is complete when $X$ is Banach space, and $Y$ is a closed subspace of $X$?
@Lukkio Set $n_1$<1. Since $(x_n)$ is Cauchy sequence, then you can find some $m$ such that $\Vert x_m-x_{n_1}\Vert<2^{-1}$. So set $n_2=m$. Again, since $(x_n)$ is a Cauchy sequence you can find $m'$ such that $\Vert x_{m'}-x_{n_2}\Vert<2^{-2}$. So set $n_2=m'$. And etc.
Jul
18
accepted Strict coisometries and operator norm.
Jul
17
revised Faulhaber Formula Identity
deleted 6 characters in body
Jul
16
comment Prove that $\displaystyle\sum_{n=1}^{\infty}a_n< \infty $ imply $\lim_{x \rightarrow \infty} \frac{1}{x} \sum_{n \leq x} n a_n=0$
Each summand tends to zero, but the count of summands is arbitrarily large, so this proof doesn't stand
Jul
11
comment $\ell_{p}$ spaces with pointwise multiplication
yes $\phantom{}$
Jul
11
answered $\ell_{p}$ spaces with pointwise multiplication
Jul
10
comment Operator norm on $M_n(A)$ by acting on $A^n$ or on $(A^+)^n$
Try the simplest case $A=\mathbb{C}$
Jul
7
revised How do I prove that $\lim_{(x,y)→(0,0)}\frac{1-\cos(x^2+y^2)}{\sqrt{x^2+y^2}} = 0$
edited tags