determine the subset Q

Question

let $C_b(\mathbb{R}^d)$ be the Banach space of bounded continuous real-valued functions on $\mathbb{R}^d$, we denote $\Vert.\Vert$ the euclidean norm on $\mathbb{R}^d$. so im trying to determine Q a subset of $C_b(\mathbb{R}^d)$, if it exists, such that $\forall x, y \in \mathbb{R}^d$ $$\Vert x-y\Vert^2=\sup_{u\in Q}\big(u(x)-u(y)\big)$$ i did prove it for $\Vert.\Vert$ but i didnt for $\Vert.\Vert^2$

Answer 1

No such set exists. The defining condition implies that any $u\in Q$ is Holder continuous with exponent $2$, and only constant maps satisfy this.