In his book Topological Function Spaces Arhangel'skii says that "it is well known that every nontrivial locally convex linear topological space $X$ is homeomorphic to a space of the form $Y \times \mathbb{R}$, for some space $Y$".
I've been looking for a proof of this result but haven't found anything, which leads me to believe that this is a standard fact that everyone knows. Could any of you please help me with a reference in which this result appears?
Best Answer
I guess that non-trivial means that the locally convex space $X$ is not endowed with trivial topology $\{\emptyset,X\}$.
This implies that $X\neq \overline{\{0\}}$ (since this closure does have the trivial topology). For $y\notin \overline{\{0\}}$, the Hahn-Banach theorem (applied to $L=\{ty: t\in\mathbb K\}$ and the linear functional $ty\mapsto t$ which is continuous because there is a continuous seminorm $p$ on $X$ with $p(y)>0$) implies that there is a continuous linear functional $\varphi\in X'$ with $\varphi(y)=1$. Then $X$ is isomorphic (in the category of topological vector spaces which is much more than homeomorphic) to $Y\times \mathbb K$ for the kernel $Y$ of $\varphi$, an isomorphism is given by $x\mapsto (x-\varphi(x)y,\varphi(x))$.
If, for complex locally convex spaces, you insist on a homeomorphy to $Y\times \mathbb R$ identify $\mathbb C=\mathbb R\times \mathbb R$.
Since this is such a simple application of Hahn-Banach, I doubt that it is worthwhile to search for an explicit reference.