Is every set a subset of a vector space?

Question

I was taught that a functional is a map from a subset (not subspace) of a vector space into the reals, $F: D\subset V \to \mathbb{R}$.

I know there are other definitions, but is there any reason to specify "subset of vector space" in this case? Couldn't one say "set" instead, or are there sets which you can't include in a vector space?

Answer 1

Yes, every set can be made into a subset of something that has a vector space structure. But that's not really the important thing here.

What the definition means is that you you say "here is a functional", then you're implicitly promising to tell me not only what the map you're thinking of, but also which vector space you're thinking of the domain as a subset of.

If you just show me the map itself, then I won't know what you're talking about -- we can then speak about its properties as a map, but we can't speak about most of its properties as a functional (for example, is it linear?) before you reveal which vector space you're thinking of.

Formally, one could insist on saying that a functional is the combination of a map $D\to\mathbb R$ with superset of $D$ and a vector space structure on that superset. That just gets too wordy for casual use, so as a matter of language the common convention is to express such things in the shorthand way you've seen.

Answer 2

A subset needn't be a subspace. For example, the single point set $\{1\}$ is a subset of $\mathbb R$, but not a subspace of it. The only subspaces of $\mathbb R$ are $\mathbb R$ and $\{0\}$.

One often needs to speak of linearity, so that often gives rise to the requirement that the domain reside within a vector space. Linearity requires things like $$F(a{\bf x} + b{\bf y}) = aF({\bf x}) + bF({\bf y})$$ The expression on the right makes sense, but the expression on the left only makes sense if there is scalar multiplication and vector addition (otherwise, what would "$a{\bf x} + b{\bf y}$" mean?).