# Norbert

 Jul 31 comment Infinite product of sine function@PhoemueX this was a sloppy definition, now fixed Jul 31 revised Infinite product of sine functionadded 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 spaceDo 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