Non-locally convex topologies on $\mathbb{R}^{n}$ compatible with the vector space structure

Question

So I know that every locally convex topology on $\mathbb{R}^{n}$ is equivalent to the norm topology. Are there any non-trivial examples of non-locally convex topologies on $\mathbb{R}^{n}$ that still make it into a topological vector space?

Answer 1

No. In Theorem 1.21 of Functional Analysis, by Rudin, it is shown that there is only one (Hausdorff) topology on $\mathbb{C}^n$ that makes it a topological vector space. The same is true for $\mathbb{R}^n$.

EDIT: For a non-Hausdorff topology $\tau$ on $\mathbb{R}^n$, let $N=\overline{\left\{0\right\}}^\tau$. We can consider it's "Hausdorffization" $\mathbb{R}^n/N$, which is a finite-dimensional Hausdorff TVS, say with a basis $x_1+N,\ldots,x_m+N$. By the result above, the quotient topology on $\mathbb{R}^n/N$ comes from the norm $\Vert\sum_{i=1}^m\lambda_i x_i+N\Vert=\max_{1\leq i\leq n}|\lambda_i|$.

Let $y_{m+1},\ldots,y_n$ be a basis for $N$, so $x_1,\ldots,x_m,y_{m+1},\ldots,y_n$ is a basis for $\mathbb{R}^n$. Let's show that $\tau$ is induced from the seminorm $$\rho\left(\sum_{i=1}^m\lambda_i x_i+\sum_{i=m+1}^n\lambda_i y_i\right)=\max_{1\leq i\leq n}|\lambda_i|.$$

If a net $\left\{\sum\lambda_i^sx_i+\sum\lambda_i^sy_i\right\}_{s\in A}$ converges to $0$ in the $\rho$-topology, then $\lambda_i^s\rightarrow 0$ for $1\leq i\leq n$, so, since the operations are continuous, $\sum\lambda_i^sx_i\rightarrow 0$ in $\tau$. Also, the topology $\tau$ restricted to $N$ is simply the indiscrete topology, so any net converges to any point. In particular, $\sum\lambda_i^sy_i\rightarrow 0$ in $\tau$. Therefore, using again the continuity of sum, $\sum\lambda_i^sx_i+\sum\lambda_i^sy_i\rightarrow 0$ in $\tau$.

Conversely, is $\sum\lambda_i^sx_i+\sum\lambda_i^sy_i\rightarrow 0$ in $\tau$, then, using the quotient mapping $\mathbb{R}^n\rightarrow\mathbb{R}^n/N$, we obtain $\sum\lambda_i^sx_i+N\rightarrow 0$, so $\lambda_i^s\rightarrow 0$ for $1\leq i\leq m$, which precisely means that $\sum\lambda_i^sx_i+\sum\lambda_i^sy_i\rightarrow 0$ in the $\rho$-topology.

That way, the topology $\tau$ is obtained in a standard manner from a seminorm.