Let $X$ be a Banach space. Let $T: X \rightarrow \ell_{\infty}$ be a bounded linear operator. Show that there is a bounded sequence $\left(f_{n}\right)_{n=1}^{\infty}$ in $X^{*}$ such that $T x=\left(f_{n} x\right)_{n=1}^{\infty}$ for all $x \in X$.

Fix $1<p<\infty$. Define the Banach space $\ell_{p}$ and briefly explain why it is separable. Show that for $x \in \ell_{p}$ there exists $f \in \ell_{p}^{*}$ such that $\|f\|=1$ and $f(x)=\|x\|_{p}$. [You may use Hölder's inequality without proof.]

Deduce that $\ell_{p}$ embeds isometrically into $\ell_{\infty}$.