Results 1 to 2 of 2

Thread: precompactness in sequence spaces

  1. #1
    Newbie
    Joined
    Nov 2010
    Posts
    19

    precompactness in sequence spaces

    I have 2 questions about precompactness in sequence spacesDoh)
    1. Is A = { $\displaystyle (x_{n} : \sum_{n=1}^{\infty} \sqrt{n} |x_{n} | \leq 1 $ } is precompact in $\displaystyle c_{0}$ = set of all sequences converging to 0 equipped with the sup norm ?
    2. Is A= { $\displaystyle (x_{n} : n x_{n} \rightarrow 0 $ as $\displaystyle n \rightarrow \infty$ } in $\displaystyle l_{1} $ ?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member girdav's Avatar
    Joined
    Jul 2009
    From
    Rouen, France
    Posts
    678
    Thanks
    32
    1. We notice that if $\displaystyle (x_n)_n\in A$ we have $\displaystyle |x_n|\leq\frac 1{\sqrt n}\, \forall n$. Given an $\displaystyle \varepsilon>0$, we can find a $\displaystyle N$ such that $\displaystyle \frac 1{\sqrt N}\leq \varepsilon$. We can conclude because $\displaystyle \left[0,1\right]^{N-1}$ is precompact.
    2. Let $\displaystyle e^{(n)}$ the sequence defined by $\displaystyle e^{(n)}_k=\begin{cases}1& \mbox{if } n=k\\
    0& \mbox{if } n\neq k
    \end{cases}$.
    We notice that $\displaystyle e^{(n)}\in A$ and if $\displaystyle m\neq n,\: ||e^{(n)}-e^{(m)}||_{l^1}=\sqrt 2$. We conclude that $\displaystyle A$ is not precompact.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Any metric spaces can be viewed as a subset of normed spaces
    Posted in the Differential Geometry Forum
    Replies: 5
    Last Post: Dec 15th 2011, 03:00 PM
  2. precompactness in L p and C[0,1]
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: Feb 27th 2011, 01:38 AM
  3. Prove that the little l(p) sequence spaces are complete.
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: Sep 30th 2010, 07:17 PM
  4. Replies: 0
    Last Post: Jul 4th 2010, 12:05 PM
  5. Replies: 3
    Last Post: Jun 1st 2008, 01:51 PM

Search Tags


/mathhelpforum @mathhelpforum