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I am stuck with the following problem:

Give example of an infinite dimensional vector space V and two norms $\theta$ and $\rho$ on V and the sequence $\{x_{n}\}_{n\geq 1}$ of V such that:

  • The sequence $\{x_{n}\}_{n\geq 1}$ is cauchy in $(V, \rho)$
  • The sequence $\{x_{n}\}_{n\geq 1}$ is not cauchy in $(V, \theta)$

It is easy to find a vector space, a norm, and a sequence that meet one of the items but I do not know how to proceed.

Any help is appreciated.

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    $\begingroup$ What kind of infinite dimensional vector spaces and norms on them do you know? $\endgroup$
    – Vercassivelaunos
    Commented Dec 10, 2021 at 18:39
  • $\begingroup$ In order for your example to be explicitly written: At least one of the norms will be not complete. $\endgroup$
    – GEdgar
    Commented Dec 10, 2021 at 19:29
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    $\begingroup$ Let $V$ be the set of every real sequence $r=(r_j)_{j\in\Bbb N}$ such that $\infty>\|r\|_1=\sum_{j\in\Bbb N}|r_j|.$ This space is called $\ell_1$ or $\ell^1$. Another norm on $V$ is $\|r\|_{\infty}=\sup_{j\in\Bbb N}|r_j|,$ which is called the $\ell_{\infty}$ (or $\ell^{\infty}$) norm. Let $r(n)=(r_{n,j})_{j\in\Bbb N}$ where $r_{n,j} =1/j$ if $j\le n$ and $r_{n,j}=0$ if $j>n$. Then $(r(n))_n$ is $\ell_{\infty}$-Cauchy but is not even $\ell_1$-bounded . $\endgroup$
    – DanielWainfleet
    Commented Dec 10, 2021 at 20:02

1 Answer 1

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Consider $V=\mathbb{R}[x]$, the space of real valued one variable polynomials. For each polynomial $f \in V$ suppose

$$||f||_{\theta}=\int_{[0,0.5]} |f(x)|dx \text{ and }||f||_{\rho}=\max(|f|)=\max\{|f(x)| ; x \in [0,1]\},$$

it is not hard to see that $\theta$ and $\rho$ are norms on $V$.

Now for the sequence $\{p_{n}\}$, where $p_{2n} = x^n$ and $p_{2n-1} = 0$, $p_{n}$ is Cauchy in $(V,\theta)$. Take $\epsilon > 0$ and $N$ such $\dfrac{2}{N} < \epsilon$, now for $n,m \geq N$ we have $||p_{n}-p_{m}||_{\theta} \leq \dfrac{1}{n}+\dfrac{1}{m} \leq \dfrac{2}{N} < \epsilon$; but it is not Cauchy in $V,\rho$ as $||x^{2n}-x^{2m-1}||_{\rho}=1$ for each $n,m$.

I think the point is to find the second norm, $\rho$, such that the the sequence you have found in the first part is not Cauchy.

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