Let $V $be a vector space. Prove/Disprove: There is a norm $\|\cdot\|$, such that all subsets of $V$ are open sets in $(V,\|\cdot\|)$.

Question

The Assignment:

Let $V$ be a vector space over $\mathbb{R}$ with $V \not= \{0\}$. Prove or disprove:

There is a norm $\|\cdot\|_d$ on $V$, such that all subsets of $V$ are open sets in $(V,\|\cdot\|)$.

I have read about topologies which are discrete, but I don't know how to define the norm such that the property is fulfilled.

Any help would be appreciated.

Answer 1

HINT: This is impossible. Show that there is a homeomorphism between a $1$-dimensional subspace and $\Bbb R$.

Answer 2

Consider $x\in V$, $x\ne0$. What can you say about the sequence $$ x_n=\frac{1}{n}x\qquad(n\ge1) $$ by considering the sequence $(\|x_n\|)_{n\ge1}$?

Answer 3

Yet another hint: All norms on $\mathbb R^n$ are equivalent (Note: It 's true only when the vector space is finite dimensional) and thus generates the same topology. Thus one need only consider the standard norm on $\mathbb R^n$ and check whether the topology is discrete.