If $\langle{x,y}\rangle = \|{x}\|_{1}\|{y}\|_{\infty}$. Determine what relationship there is between $x$ and $y$. I have the following problem:
Let $a_{1}, \ldots a_{n},b_{1},\ldots b_{n} \geq 0$, 
$x = (a_{1}, \ldots a_{n})$, y=$(b_{1},\ldots b_{n})$. If 
$\langle{x,y}\rangle = \|{x}\|_{1}\|{y}\|_{\infty}$
Determine what relationship there is between $x$ and $y$.
My attempt:
From the hypotheses I get that:
$\sum_{k=1}^{n}a_{k}b_{k} = \sum_{k=1}^{n}a_{k}\|{y}\|_{\infty} $
The answer is that $y $ is a constant vector?
 A: The case where $y$ is the zero vector is direct. Otherwise, from your equation we have, equivalently (why?), $$\sum_n a_k \left(b_k/\|y\|_\infty\right)=\sum_n a_k.$$
Since $0 \leq b_k \leq \|y\|_\infty$ with equality only at the indices where the maximum value occurs (call them $i \in [n]$), then the conclusion is that $a_k=0$ at all other indices $j \not = i, j \in [n],$ for if this fails for any such $j$, then $\frac{b_j}{\|y\|_\infty}<1$ and we would then have $$\sum_j a_k \left(b_k/\|y\|_\infty\right)<\sum_j a_k,$$ and hence $$\sum_n a_k \left (b_k/\|y\|_\infty\right)=\sum_j a_k \left(b_k/\|y\|_\infty\right)+\sum_ia_k<\sum_n a_k, $$ contradicting the above equation equivalent to the assumption in the problem. 
A: If all $a_i$ and all $b_i$ are non-negative, then: $b_i\le\|y\|_\infty$ so $\sum_{i=1}^na_ib_i\le \sum_{i=1}^na_i\|y\|_\infty=\|x\|_1\|y\|_\infty$. 
For the equality to hold, all the equalities $a_ib_i=a_i\|y\|_\infty$ must hold. Thus, for all $i$ we have $a_i\ne 0 \implies b_i=\|y\|_\infty=\max_{j=1,\ldots\,n}b_j$.
Let us define $\mathbf 1=(1,1,\ldots 1)$. With the additional condition such as $a_i\gt 0$, it would be easy to conclude that $b_i=\|y\|_\infty$ for all $i$, i.e. that $y=\|y\|_\infty\mathbf{1}$.
Without such a condition, there are all sorts of examples, e.g. $x=(1,0), y=(2, 1)$. All I can say is: $\langle x,\|y\|_\infty\mathbf{1}-y\rangle=0$ or $x\bot\|y\|_\infty\mathbf{1}-y$.
Worth checking here whether there is a misprint in the book and the author actually wanted to say $a_i\gt 0$. (Any errata provided?)
A: Since $\sum_k (\|y\|_\infty |x_k| - x_k y_k) = 0$ and $\|y\|_\infty |x_k| \ge x_k y_k $, you must have
$\|y\|_\infty |x_k| = x_k y_k$ for all $k$.
Since $x_k, y_k \ge 0$ we see that if $x_k >0$ then $y_k = \|y\|_\infty$ otherwise $y_k \le \|y\|_\infty$.
