In Axler's Linear Algebra Done Right, he proves that statement in the title however only addresses finite-dimensional vector spaces in the proof.
My question: Does it also apply to infinite-dimensional vector spaces?
His proof:
My confusing arrises as the statement is the part: "every subspace of V is a direct…" To me, it seems V is just an arbitrary vector space that can be finite or infinite dimensional, however he only proves it for finite-dimensional spaces.
Please note that this is relatively early on in the book so I feel like if it is true for infinite-dimensional vector spaces, we don't know enough yet which is why he (Axler) doesn't prove it for them as well.
Best Answer
Because that proof uses the fact that every finite-dimensional vector space has a basis. However, the statement “every vector space has a basis” neither can be proved to be true nor it can be proved to be false; it depends upon which set theory you are using. Also, that proof from Axlers's textbook uses the fact that every basis of a subspace can be extended to a basis of the whole space, assuming that the whole space is finite-dimensional. Again, this statement neither can be proved to be true nor it can be proved to be false.
However, in the specific case in which your set theory is ZFC (Zermelo-Fraenkel theory with the axiom of choice), then both statements can be proved to be true.