The problem: Let $p\in(0,\infty)$ and $(E_n)_{n\in\Bbb N}$ a sequence of Banach Spaces. If we define $$\Bigl(\sum E_n\Bigr)_p:=\biggl\{ (x_n)_{n\in\Bbb N}~|~x_n\in E_n, \forall n\in\Bbb N, ~\text{and}~\|(x_n)_{n\in\Bbb N}\|_p:=\biggl(\sum_{n\in\Bbb N}\|x_n\|_{E_n}^p\biggr)^{\frac{1}{p}}<\inft...

I'm having trouble in proving that the following space is reflexive: $$E = \{ x= (x_n) : x_n \in \mathbb{R}^n \text{ and } \sum \|x_n\|^2_\infty < \infty\}$$ with the norm $$ \|x\| = (\sum \|x_n\|^2_\infty)^\frac{1}{2}$$ I already tried an analogous of the proof that the $l_p$ spaces are refl...

Let $\varphi\in \ell'_p$ and $\{e_n\}$ the standard Schauder basis of $\ell_p$, and $y_n=\varphi(e_n)$. We shall show that $y=(y_1,\ldots,y_n,\ldots)\in \ell_q$ and $\varphi(x)=\langle x,y\rangle$, for all $x\in\ell_p$. Let first $\varphi_n(x)=\sum_{k=1}^n x_ky_k$, where $x=(x_1,\ldots,x_n,\ldot...

Assume $x_n^k, x_n$ to be real for all $k$ and for all $n$. Suppose that I have $x_k^{(n)}\to x_k$ as $n\to \infty$ for every $k\in \mathbb N$. Then can something be said about $\sup_k|x_k^{(n)}-x_k|$? Given any $\epsilon>0$, $(\forall k\in \mathbb N) \exists N_k\in \mathbb N, n\ge N_k\implies |x...

Exercise: One evening Dr. Matrix discovered some formulas that might even be classed as more remarkable than those of exercise 20: $$\frac{1}{(a-b)(a-c)} + \frac{1}{(b-a)(b-c)} + \frac{1}{(c-a)(c-b)}=0$$ $$\frac{a}{(a-b)(a-c)} + \frac{b}{(b-a)(b-c)} + \frac{c}{(c-a)(c-b)}=0$$ $$\frac{a^2}{(a...