# Example of a sequence on an infinite-dimensional vector space with respect to different norms [closed]

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.

• What kind of infinite dimensional vector spaces and norms on them do you know? Commented Dec 10, 2021 at 18:39
• In order for your example to be explicitly written: At least one of the norms will be not complete. Commented Dec 10, 2021 at 19:29
• 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 . Commented Dec 10, 2021 at 20:02

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.