Are $\ell_p$ spaces Banach algebras under pointwise multiplication?

Question

Is $\ell_{p} $, the set of all scalar $p $-summable sequences, a Banach algebra with pointwise multiplication and the $p$-norm ? (We may assume that we are in the case of real scalars.)

Answer 1

It is, what have you tried? Take two sequences $x,y\in \ell_p$. Then

$$\sum_{n=1}^\infty |x_ny_n|^p \leqslant \sum_{n=1}^\infty |x_n|^p\|y\|^p_{\ell_\infty}=\|x\|^p\|y\|^p_{\ell_\infty}\leqslant \|x\|^p\|y\|^p.$$