Understanding the Vector Space Structure of Complex Numbers on Real Numbers

When delving into the fascinating world of vector spaces and group theory, one cannot help but marvel at the intricacies and nuances that arise from different underlying set theories. In this exploration, we will discuss the vector space structure of complex numbers over real numbers, and how this structure changes based on different axiomatic assumptions.

Vector Spaces and Abelian Groups

In mathematics, a vector space is a structure comprising a set of vectors along with operations of addition and scalar multiplication. An abelian group, on the other hand, is a set equipped with an associative binary operation and an identity element, such that the operation is commutative and every element has an inverse.

When we consider the set of real numbers, (mathbb{R}), we can define multiple vector space structures on it. The key differences arise when we apply the Axiom of Choice (AC) or other set-theoretic assumptions.

The Role of the Axiom of Choice

The Axiom of Choice states that for any collection of non-empty sets, there exists a choice function that selects one element from each set. In the context of vector spaces over the real numbers, AC allows us to construct a basis for the real numbers over the rationals, establishing that (mathbb{R}) and (mathbb{R}^2) are isomorphic as abelian groups. Let's explore why this is significant:

Under the Axiom of Choice, we know that (mathbb{R}) and (mathbb{R}^2) can be isomorphic as abelian groups. This is a stunning and seemingly counterintuitive result because the dimensions of these vector spaces are different ((mathbb{R}) has dimension 1 while (mathbb{R}^2) has dimension 2). Given this isomorphism, we can use it to construct an alternative scalar multiplication on complex numbers, (mathbb{C}). Specifically, let (phi: mathbb{C} to mathbb{R}) be a group isomorphism, and define a new scalar multiplication on (mathbb{C}) by

(a cdot z phi^{-1}(aphi(z))), for all (a in mathbb{R}) and (z in mathbb{C}).

This redefining of scalar multiplication makes (mathbb{C}) a real vector space isomorphic to (mathbb{R}), rather than (mathbb{R}^2).

Implications of Different Set Theories

While most mathematicians accept the Axiom of Choice due to its wide acceptance and consistency with everyday mathematical practices, there are models of set theory where the Axiom is not assumed. In such models, the vector space isomorphism between (mathbb{R}) and (mathbb{R}^2) may not hold.

In the Solovay universe where every subset of the reals is measurable, (mathbb{C}) has only the obvious vector space structure. Any group homomorphism (f: mathbb{C} to mathbb{C}) is linear with respect to the canonical vector space structure. Suppose (f) is a non-trivial homomorphism and consider a line passing through (0) that is not completely in the kernel of (f). Identifying (mathbb{R}) with this line, we can project the image to (mathbb{R}). This projection provides a solution to the Cauchy functional equation (g(x y) g(x)g(y)). Every solution of this equation is either linear or not measurable. If all sets are measurable, the latter is impossible.

The fact that most mathematicians do not delve into set theory but prefer to work with a consistent and widely accepted framework highlights the practical significance of such models. The Zermelo-Fraenkel set theory with the Axiom of Choice (ZFC) is the most commonly used due to its consistency and ease in problem-solving.

Conclusion

The isomorphism between (mathbb{R}) and (mathbb{R}^2) as abelian groups, provided by the Axiom of Choice, is a remarkable and somewhat unsettling result. While this result is often taken for granted in many mathematical contexts, it underscores the importance of foundational assumptions in set theory. For those who wish to avoid set-theoretic nuances, a practical rule is: if a statement holds in a model of ZF, it holds for every concrete example.

Understanding these concepts helps mathematicians navigate the complexities of vector spaces while maintaining a solid theoretical foundation.