I'm learning some classical geometry and I have a few questions redarding some topological algebra facts.
First, I would like to prove that if $\tau$ and $\tau'$ are two Hausdorff topologies on $V$ (finite dimensional) that make it into a topological vector space (operations are continuous), then they are equal. I tried approaching it in many different ways but they all turned out to be useless in some way or another.
My second problem is in the proof that $T_p\mathbb{S}(V) = \text{Lin}(\mathbb{R}v, V/\mathbb{R}v)$. I really don't care about the formal proof, I can see the details later, but I'm trying to build some intuition and thus it came to my mind a sort of conjecture and I have NO idea if it's true
If $V$ is a finite dimensional vector space, then it has a inner product $\langle \cdot, \cdot \rangle$ and we can see that $V/\mathbb{R}v = v^\perp$. Fix $v \in V$. Is it true that every linear transformation $\phi \colon V \to V$ satisfying $f(\mathbb{R}v) \subset v^\perp$ is a rotation composed with product by some scalar?
I would really appreciate some help regardin these two problems. Just to explain my motives: if the above conjecture is true, then we can think of the isomorfism $T_v\mathbb{S}(V) = \text{Lin}(\mathbb{R}v, V/\mathbb{R}v) = \text{Lin}(\mathbb{R}v, v^\perp)$ as the following: take some vector $u \in T_v\mathbb{S}(V)$ and translate it to $v^\perp$ (here we need to translate it because $v^\perp$ and $T_v\mathbb{S}(V)$ are parallel hyperplanes), starting at the origin. Then, take the map that rotates and scales $\mathbb{R}v$ so that $v$ is taken to $u$. This is linear because it is a rotation and a multiplication by some scalar. The same argument can be done backwards, so that we have a correspondence.