Is $\tau_1$ is stronger than $\tau_2$?

Question

let norms $||x||_1 = \sum_{i=1}^{n} |K_i|$ and $ ||x||_2= (\sum_{i=1}^{n}| K_i|^2)^{\frac{1}{2}}$ induce a topology $\tau_1$ and $\tau_2$ on $\mathbb{R}^n $ ,the $n$ dimensional euclidean space ,then

choose the correct option

$a)$ $\tau_1$ is weaker than $\tau_2$

$b)$ $\tau_1$ is stronger than $\tau_2$

$c)$ $\tau_1$ is equivalent $\tau_2$

$d)$ $\tau_1$ and $\tau_2$ are incomparable

My attempts : I know that $||x||_2 \le ||x||_1 $ this implies $\tau_1$ is stronger than $\tau_2$ so the correct answer will be option b)

is it corrcet ??

Any hints/solution will be appreciated

thanks u

Answer 1

From $\|x\|_2\leq \|x\|_1\leq \sqrt n \|x\|_2$ we see c) is correct.