It's always interesting how properties of finite dimensional objects translate (or not translate) to infinite dimension.
One can prove that for finite dimensional vector space $V$ set $\emptyset \subsetneq H \subset V$ is a hyperplane iff $H = f^{-1}(a)$ for some non-zero linear functional $f: V \rightarrow \mathbb{R}$ and some $a \in Im(f)$.
The above equivalence allows us to define term hyperplane for vector spaces of arbitrary dimension as preimages of linear functionals.
If $H = f^{-1}(a)$ and $f$ is continuous it's easy to prove that (whatever dimension space $V$ has) $H$ as well as closed half-spaces $$ H_-(f) = \{x \in V ~~|~~ f(x) \leq a\}$$ $$ H_+(f) = \{x \in V ~~|~~ f(x) \geq a\}$$ are closed sets and open half-spaces $$ H_{-}^{o}(f) = \{x \in V ~~|~~ f(x) < a\}$$ $$ H_{+}^{o}(f) = \{x \in V ~~|~~ f(x) > a\}$$ are open sets. Particularly this properties hold for finite-dimensional vector spaces since in this case any linear functional is continuous.
So here's my question: do this topological properties still hold in infinite dimensional spaces if we do not require the functional to be continuous? My intuition says "no" but i didn't succeed in finding counter-examples.