> [!definition] > > Let $E$ be a [[Banach Space|Banach space]] over $\real$ and $\mu \in E^*$ be a [[Bounded Linear Functional|bounded linear functional]]. Define > $ > E^\pm_\mu = \bracs{x \in E: \pm\mu(x) \ge 0} \quad E^0_\mu = \ker (\mu) > $ > If $\mu \ne 0$, then $E_\mu^\pm$ are the $\mu$-**half-spaces** of $E$, which inherits the subspace topology. If $\mu = 0$, then $E_\mu^\pm = E_\mu^0 = E$. > > Suppose that $\mu \ne 0$, then since $\ker(\mu)$ is co-finite dimensional, it [[Split Subspace|splits]] $E$ as $\ker(\mu) \oplus \real z$ for some $z \in E$ with $\mu(z) = 1$. If $\nu \in E^*$ and $\ker(\mu) = \ker(\nu)$, then $\nu = \lambda \mu$ for some $\lambda \in \real$.