Estimation with inner products

Question

Let $x$, $u$, $w$ be vectors in $\mathbb{R}^n$ suppose that $u$, $w$ are close: $$ \| u - w \| < \frac{1}{n}, $$ and we have an upper bound for the inner product: $$ \langle x, u \rangle < 1. $$ Is it possible to find an upper bound for the inner product $\langle x, w \rangle$ without using any additional information about $x$, for example $\|x\|$?

Answer 1

Choose some $u \neq 0$ and $\delta \bot u$ with $\|\delta\| = {1 \over 2n}$ and let $w=u+\delta$, then $\|w-u\| < { 1 \over n}$.

Let $x(t) = {1 \over 2 \|u\|^2} u + t \delta$, note that $\langle x(t), u \rangle = {1 \over 2}$ for all $t$.

However, $\langle x(t), w \rangle = { 1 \over 2} +t {1 \over (2n)^2}$.

Hence you cannot bound $\langle x, w \rangle$ without more conditions.