Why is $|x · y| ≤ ||x||_1||y||_∞$?

Question

So let $||x||_∞ := $ max $_{j=1,...,n}|x_j|$ and by Cauchy-Schwarz, $|x · y| ≤ ||x||_2||y||_2$ .

Why then does $|x · y| ≤ ||x||_1||y||_∞$ ?

I'm not sure how to show this.

Answer 1

You know that $\lvert x_i\rvert \le \max_{j=1,\dotsc,n}\lvert x_j\rvert$ for each $i=1,\dotsc, n$. Hence $\lvert x_i\rvert\le \lVert x\rVert_\infty$ for each $i=1,\dotsc,n$. Now, \begin{align*} \lvert x\cdot y\rvert &= \left\lvert\sum_{i=1}^nx_iy_i\right\rvert \le \sum_{i=1}^n \lvert x_i\rvert\cdot \lvert y_i\rvert\\ &\le \sum_{i=1}^n\lVert x\rVert_\infty\cdot \lvert y_i\rvert\\ &= \lVert x\rVert_\infty\cdot \sum_{i=1}^n\lvert y_i\rvert = \lVert x\rVert_\infty\cdot \lVert y\rVert_1. \end{align*}

Answer 2

$$|x\cdot y| \le \sum_{i=1}^n |x_i\bar y_i|\le \sum_{i}\left((\sup_j|x_j||y_i|)\right)=\sup_j|x_j|\sum_i|y_i|=\|x\|_{\infty}\|y\|_1$$