Vector space of functions on an empty domain?

Question

I know that these sorts of pathological cases are irrelevant, but I want to "practice" as it were and so want to understand the following. Hoffman and Kunze give the following example of a vector space:

The space of functions from a set to a field. Let $F$ be any field and let $S$ be any non-empty set. Let $V$ be the set of all functions from the set $S$ into $F$.

They give the usual definitions of addition and scalar multiplication. At any rate, my question is about why they give the nonempty requirement. If $S$ is empty, then $V$ itself is not empty but rather contains the empty set (the empty set can be defined as a function on an empty set), $V = \{ \emptyset \}$.

If I define $\emptyset$ as my neutral element, then why on earth can't this be a vector space?

Answer 1

It is a vector space (a zero dimensional/trivial vector space).

There are some people who don't like functions with empty domains (they introduce a few issues; e.g., it is not true that "a function is injective if and only if it has a left inverse" if you allow functions with empty domain). Or who find them somewhat confusing. So they don't want to deal with them.

I suspect that is what is at issue here. The author does not want to consider functions with empty domain, so they exclude $S=\varnothing$. The wording of the example does not categorically state that you will not get a vector space when $S$ is empty: it just does not treat that case and is silent about it.

You are correct that a one element set, say $\{\star\}$, is a vector space (over any field): the trivial vector space with operations $\star+\star = \star$ and $\alpha\cdot\star = \star$ for any scalar $\alpha$. Perfectly fine vector space. I know of no one who would exclude this vector space (it would introduce all sorts of exception statements in standard theorems: for example, it would no longer be true that an intersection of subspaces of $V$ is again a subspace of $V$, we would have to add a clause stating that the intersection consists of more than the zero vector).

So the problem is not with the one element vector space. The issue almost certainly resides with the author's desire to avoid functions with empty domain, not with the resulting object.