Proving Holders inequality for the sequence space $l_p (\mathbb(N)$

Question

We first look at when $p=1$ and $q=\infty$

And we look at the non trivial case when the sequences $x=(x_k)_{k \in \mathbb{N}}$ and $y=(y_k)_{k \in \mathbb{N}}$ are both not equal to zero.

We first start of with the estimate $|x_k y_k| \leq |x_k| |y_k| \leq |x_k| sup_{m \in \mathbb{N}} |y_m| = |x_k ||y||_\infty$

Where does this estimate come from?

How can we say that if $x=(x_k)_{k \in \mathbb{N}} \in l_1 (\mathbb{N})$ than $(|x_k| ||y||_\infty)_{k \in \mathbb{N}} \in l_1(\mathbb{N})$?

Answer 1

$|x_k y_k| = |x_k| |y_k|$ is a basic fact of absolute value / modulus.

You can upper bound this by replacing $|y_k|$ with an upper bound on $|y_k|$: $|y_k| \leq \sup_m |y_m| = \lVert y \rVert_\infty$.

Thus, $|x_k y_k |\leq |x_k| \lVert y \rVert_{\infty}$.

As for $|x_k| \lVert y \rVert_{\infty}$ being in $\ell^1$, $\sum_k |x_k| \lVert y \rVert_{\infty} = \lVert y \rVert_{\infty} \sum_k |x_k| = \lVert y \rVert_{\infty} \lVert x \rVert_1 < \infty$ by assumption of $y$ being in $\ell^\infty$ and $x$ being in $\ell^1$.