Redimensioning of Euclidean Spaces

TL;DR

This paper demonstrates that the dimension of Euclidean spaces over the reals can be redefined through alternative vector operations, leveraging Cantor's result that all real spaces are bijectively equivalent, and extends this to all finite-dimensional vector spaces.

Contribution

It shows that Euclidean spaces can be restructured to have any desired finite dimension by redefining their vector operations, based on the equivalence of real spaces.

Findings

Reconstructed Euclidean spaces with arbitrary finite dimensions.

Utilized Cantor's result on the equivalence of nd imensional real spaces.

Extended the dimension redefinition to all finite-dimensional vector 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.