I we are given a vector space $(V,+,\cdot)$ over a field $\mathbb K$ (where $\mathbb K=\mathbb R$ or $\mathbb K=\mathbb C$), is there the finest topology $\mathcal T$, such that $(V,\mathcal T)$ is a topological vector space?
I'll explain below what I have tried. This seems a very natural question to me, so I guess the answer (whether it is affirmative or negative) should be well known, at least to some local experts in topological vector spaces.
First guess would be the discrete topology, but discrete topology does not give a topological vector space. Indeed, if $\cdot \colon \mathbb K\times V\to V$ is continuous then for each fixed $v\in V$ the map $\alpha \mapsto \alpha\cdot v$ from $\mathbb K$ to $V$ should be continuous. Since the map is injective for $v\ne0$, this would imply that $\mathbb K$ has discrete topology too.
Perhaps we could get a topology such that the maps $\alpha \mapsto \alpha\cdot u$, $v \mapsto \beta\cdot v$, $v\mapsto v+u$ are continuous for each $u\in V$ and $\beta\in\mathbb K$ by iteratively taking the initial topology w.r.t. these maps, starting from the discrete topology on $V$ and the usual topology on $\mathbb K$ and iterating this proces. But even if this worked, we would get only separate continuity for $\cdot \colon \mathbb K\times V\to V$, and we want this map to be jointly continuous.
While looking for answer to my question, I found out about the existence of finest locally convex topology (Google Books), which is mentioned, for example in this question or this question. This is different from what I am asking here.
I would say that there is a finest vector space topology. However, my argument may be completely wrong...
Let $(\tau_i)_{i\in I}$ be the family of all vector space topologies on $V$. Call a set $\mathcal E\subset V$ $elementary$ if it can be written as $\mathcal E=\mathcal U_{i_1}\cap \cdots \cap \mathcal U_{i_N}$, where $\mathcal U_{i_k}\in\tau_{i_k}$. Finally, let $\tau$ be the family of all subsets of $V$ which are unions of elementary sets.
Since the family of elementary sets contains $\emptyset$ and $V$ and is stable under finite intersections, it is rather clear that $\tau$ is a topology on $V$. Obviously, $\tau$ is finer than all $\tau_i$; so it remains to check that $\tau$ is a vector space topology.
Take $x,x'\in V$ and let $\mathcal W$ be a $\tau$-neighbourhood of $x+x'$. Choose an elementary set $\mathcal E=\mathcal U_{i_1}\cap \cdots \cap \mathcal U_{i_N}$ such that $x+x'\in \mathcal E\subset\mathcal W$. Since each $\tau_{i_k}$ is a vector space topology, one can choose $\mathcal V_k, \mathcal V'_k\in\tau_k$ such that $x\in\mathcal V_k$, $x'\in\mathcal V'_k$ and $\mathcal V_k+\mathcal V'_k\subset \mathcal U_{i_k}$. Then $\mathcal V=\bigcap_k \mathcal V_k$ and $\mathcal V'=\bigcap_k\mathcal V'_k$ are $\tau$-neighbourhoods of $x$ and $x'$ such that $\mathcal V+\mathcal V'\subset\mathcal E\subset \mathcal W$. This shows that addition is jointly continuous on $(V,\tau)\times (V,\tau)$.
Continuity of scalar multiplication can be checked in the same way.
Please let me know if this seems correct!