Generic topology on a vector space?
For a (possibly infinite-dimensional) vector space $V$, I thought about the following topology $\tau$: Let $O \in \tau$ if every $x \in O$ has the property that for every $v \in V$, there is an $\epsilon > 0$ such that $x + \alpha v \in O$ for every $\alpha$ with $| \alpha | < \epsilon$. This gives a topology indeed (right?). I think that in finite-dimensional vector spaces, this gives the standard topology (right?).
What about infinite-dimensional vector spaces? Is there a name for this topology? To me, this looks like a very generic way to define a topology on a vector space, but I've never seen this definition of a topology. Is there any connection to locally convex topologies? (I'm not very familiar with this concept) Is this topology important in any application?
Solution 1:
This is not an answer. Rather, let me explain another canonical way of endowing vector spaces with a topology.
Let $V$ be a vector space over the real numbers $\mathbf{R}$. (You can replace $\mathbf{R}$ by any topological field in everything that follows, although for the properties at the end you'll need your topological field to be complete of characteristic zero and Hausdorff.)
Suppose that $V$ is finite-dimensional. Any isomorphism $V\cong \mathbf{R}^n$ endows $V$ with a topology compatible with the vector space structure, where we endow $\mathbf{R}^n$ with the product topology. (Such an isomorphism boils down to the choice of a basis for $V$.) It is easy to see that this topology is independent of the chosen isomorphism.
Now, in general, if $V$ is not finite-dimensional, you can endow $V$ with the "inductive limit topology". This topology is characterized by the following property:
A map $f:V\to A$ is continuous if and only if its restriction $f|_{W}:W\to A$ to every finite-dimensional subspace $W$ of $V$ is continuous.
It is a direct consequence of the definitions that the inclusion $W\to V$ of a finite-dimensional subspace $W$ of $V$ is continuous. More generally, any $\mathbf{R}$-linear map $V_1\to V_2$ of vector spaces endowed with the inductive limit topology is continuous.
Properties:
Any vector space of $V$ is closed.
The topology is Hausdorff.
Let $K\subset V$ be compact. Then $K$ is contained in a finite-dimensional subspace of $V$.
The first two properties are not so hard to prove. In fact, to prove that $V$ is Hausdorff proceed as follows. Let $x$ and $y$ be in $V$. Consider the subspace $W$ generated by $x$ and $y$. Choosing a basis for $V$ gives you a projection $p:V\to W$. This projection is continuous. Let $U_x$ be an open of $W$ containing $x$ and $U_y$ an open of $W$ containing $y$ such that $U_x\cap U_y$ is empty. (Here you use that the base field is Hausdorff.) Then $p^{-1}(U_x)$ and $p^{-1}(U_y)$ separate $x$ and $y$ in $V$.
The third one is a bit trickier. You'll need to use that the base field is complete and of characteristic zero.