Vector space span by one-to-one function Consider $f$ be a one-to-one function from $\Bbb R$ to $\Bbb R.$ the question

Can $f$ form a basis?

if so , $V=\text{span}_{\Bbb Q}(f)$ would be $1$-dimensional vector space. Notice that basis does not contain $0.$ So, we can take $f(x)=e^x$, from example. Is that right ?

Yes. More generally, any nonzero singleton in any vector space forms a basis of some subspace.
