Let $$A = \Big\{(a\_1,a\_2,\dots)\ \Big|\ a\_i\ge 0, \sum\_{i=1}^\infty a\_i=1\Big\},$$ $$v(x)=\sup\left(\bigg\{\sum\_{i=1}^\infty a\_ib\_i\ \bigg|\ (a\_i)\_{i=1}^\infty,\, (b\_i)\_{i=1}^\infty \in A,\,\sup\limits\_{i\in \mathbf N} a\_ib\_i =x\bigg\}\right).$$ Certainly $v$ is an increasing function. Is $v(x)$ finite for every $x$? Is it achievable? Does $v(x)\rightarrow 0$ as $x\rightarrow 0$?



Inner Product of Given Sum Positive Sequence
We have the following inequality:


\begin{align\*}\sum\_{i =1}^\infty a\_i b\_i & \le \left(\sum\_{i = 1}^\infty a\_i^2 b\_i\right)^{1/2} \left(\sum\_{i = 1}^{\infty} b\_i\right)^{1/2} \\ & \le \left(\sum\_{i = 1}^\infty a\_i a\_i b\_i\right)^{1/2} \\ & \le \left((\sup\_i a\_i b\_i) \sum\_{i =1}^\infty a\_i\right)^{1/2}\\ & \le \sqrt{x}.\end{align\*}


Hence $\nu(x) \to 0$ as $x \to 0$.


