When I was Studying studying Banach spaces, I was confused with the following: We know that, in any Banach Space $V$, the closed unit ball is compact in the topology generated by the norm if, and only if, the dimension of $V$ is finite. But thinking about $\mathbb R$ as a vector space over $\mathbb Q$, we have a infinite dimensional an infinite-dimensional vector space and which is complete in the norm (given by the modulus) but the closed unit ball is, of course, compact in topology generated by the norm.
I took some time to discover that my mistake it was that I thought about $\mathbb R$ over $\mathbb Q$ as a Banach space. In fact, this vector space it is a complete metric space (in the sense of Cauchy sequences), but I realized latter later that the word Banach space is reserved only for vector spaces defined over the fields $\mathbb R$ or $\mathbb C$.
