Are 2 norm and infinity norm equivalent in an infinite dimensional vector space space

Question

I am trying to answer the following question:

Are the 2-norm and the infinity norm equivalent in $l^2$, the space of real valued sequences which are square summable?

($l^2 := \big\{ x = (x_i)_{i \in \mathbb{N}} \ \ \big| \ \ \sum_{i=1}^{\infty} |x_i|^2 < +\infty \big\}$)

I am aware that in an n-dimensional vector space, we are able to obtain the following inequality (whose lower bound still stands for an infinite dimensional space):

$\|x\|_\infty \le \|x\|_2 \le \sqrt{n} \|x\|_\infty$

But I am unsure of how to proceed. Are there any obvious sequences to use as counterexamples if the norms are not equivalent? Otherwise can someone give me a hint for how I can find a constant to replace $\sqrt{n}$ in the upper bound? Thanks!

Answer 1

Take $x_n = (1,1,1,...,\underset{n-th}{1}, 0,0,....)\in \ell^2.$ Then $$||x_n ||_{\infty} =1$$ but $$||x_n||_2 =\sqrt{n}$$ so these norms can't be equivalent on $\ell^2$