# precompactness in sequence spaces

• Feb 26th 2011, 03:57 PM
hermanni
precompactness in sequence spaces
I have 2 questions about precompactness in sequence spaces:(Doh)
1. Is A = { $(x_{n} : \sum_{n=1}^{\infty} \sqrt{n} |x_{n} | \leq 1$ } is precompact in $c_{0}$ = set of all sequences converging to 0 equipped with the sup norm ?
2. Is A= { $(x_{n} : n x_{n} \rightarrow 0$ as $n \rightarrow \infty$ } in $l_{1}$ ?(Shake)
• Feb 28th 2011, 11:44 AM
girdav
1. We notice that if $(x_n)_n\in A$ we have $|x_n|\leq\frac 1{\sqrt n}\, \forall n$. Given an $\varepsilon>0$, we can find a $N$ such that $\frac 1{\sqrt N}\leq \varepsilon$. We can conclude because $\left[0,1\right]^{N-1}$ is precompact.
2. Let $e^{(n)}$ the sequence defined by $e^{(n)}_k=\begin{cases}1& \mbox{if } n=k\\
0& \mbox{if } n\neq k
\end{cases}$
.
We notice that $e^{(n)}\in A$ and if $m\neq n,\: ||e^{(n)}-e^{(m)}||_{l^1}=\sqrt 2$. We conclude that $A$ is not precompact.