For $x,y \in \mathbb R^n$ let $$ d_p(x,y) = \left(\sum_{i=1}^n \def\abs#1{\left|#1\right|}\abs{x_i - y_i}^p\right)^{1/p}$$ for $1 \le p < \infty$ and $$ d_\infty(x,y) = \max\{\abs{x_i -y_i} \mid i = 1, \ldots, n\} $$ Let $B_p = \{x\in \def\R{\mathbb R}\R^n \mid d_p(x,0) < 1\}$, $1 \le p \le \infty$. Which of the following are correct?
- $B_1$ is open in the $d_\infty$-metric.
- $B_2$ is open in the $d_\infty$-metric.
- $B_1$ is not open in the $d_2$-metric.
- $B_2$ is not open in the $d_2$-metric.
Help me with the geometrical approach to solve this. How to construct those sets and all those ?? and explain the ruled out options in detail
Prove that for any $p,q$ the ball $B_p$ is contained in some rescaling of the ball $B_q$. As a consequence all the topologies are equal each other hence statements like "$X$ is open with respect to $d_p$" do not depend on $p$.