Manipulating a sum and showing it has an upper bound

Question

Let $x_1, \ldots ,x_n$ and $y_1, \ldots , y_n$ for $n \in \mathbb{N}$ be complex numbers such that $$\left| \sum_{k=1}^n x_k y_k \right| \leq 1. $$ Further I know that $$ \sum_{k = 1}^n |x_k|^2 \leq 1.$$ My question is: does $ \sum_{k = 1}^n |y_k|^2 \leq 1. $ also hold? And assuming $ \sum_{k = 1}^n |x_k|^2 \leq 1 $ and $ \sum_{k = 1}^n |y_k|^2 \leq 1 $ hold does $\left| \sum_{k=1}^n x_k y_k \right| \leq 1 $ hold? How do I proof this?

Answer 1

Consider $x_1=1$, $x_2=0$ and $y_1=0$, $y_2=\lambda$. Then $$ \left|\sum_{k=1}^2x_ky_k\right|=0\le1 $$ and $$ \sum_{k=1}^2x_k^2=1 $$ However, $$ \sum_{k=1}^2y_k^2=\lambda^2 $$ which can be as large as we want.

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The second part follows from the Cauchy-Schwarz Inequality.