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 suggested edit on 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
19
comment Noncommutative analogs of classical Banach geometric properties
@BillJohnson, thank you! This paper mostly deals with the case $1<p<\infty$. As for the edge cases, do you know a non-commutative analog of Dunford-Pettis property, a kind of property shared by bounded and nuclear operators?
Jul
18
accepted Strict coisometries and operator norm.
Jul
18
revised Noncommutative analogs of classical Banach geometric properties
edited title
Jul
18
asked Noncommutative analogs of classical Banach geometric properties
Jul
17
revised Faulhaber Formula Identity
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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
14
awarded Nice Question
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