Redimensioning of Euclidean Spaces

Abstract

A vector space over a field $\mathbb{F}$ is a set $V$ together with two binary operations, called vector addition and scalar multiplication. It is standard practice to think of a Euclidean space $\mathbb{R}^n$ as an $n$-dimensional real coordinate space i.e. the space of all $n$-tuples of real numbers ($R^n$), with vector operations defined using real addition and multiplication coordinate-wise. A natural question which arises is if it is possible to redefine vector operations on the space in such a way that it acquires some other dimension, say $k$ (over the same field i.e., $\mathbb{R}$). In this paper, we answer the question in the affirmative, for all $k\in\mathbb{N}$. We achieve the required dimension by `dragging' the structure of a standard $k$-dimensional Euclidean space (\R{k}) on the $n$-tuple of real numbers ($R^n$). At the heart of the argument is Cantor's counterintuitive result that $\mathbb{R}$ is numerically equivalent to $\mathbb{R}^n$ for all $n\in\mathbb{N}$, which can be proved through an elegant construction. Finally, we generalize the result to all finite dimensional vector spaces.