In Sheldon Axler's wonderful book Linear Algebra Done Right, one exercise goes like this: Is a subspace of the complex vector space ? My first naive guess was yes, since make up a vector space and since is a subset of . This guess turned out to be wrong due to a subtle but important detail. Here I'd like to offer a proper answer to this question.
The key to answering this question for me was to recall that a vector space is actually defined by a set and a field . Usually we refer to the typical real vector space by just and implicitly assume that . To be more precise, let denote the vector space where elements can be scaled by a scalar . With this notation we can denote the typical real vector space by .
With this clarification in mind, the question mentioned above seems ill-defined: There is no mention of what fields to use. When referring to the complex vector space we usually mean . So with the assumption that , a more precise version of the question could be:
Is a subspace of ?
With this formulation the question is much easier to reason about. In order for to be a valid vector space it needs to be closed under scalar multiplication. Now that we have made clear that it is fairly easy to see that this is not the case. Suppose and then . So is not a vector space and therefore cannot be a subspace of .
The important insight here is that whenever you deal with a mix of real and complex vector spaces, you have to be careful about what "scalar" means, i.e. what field is used.