What is $\Bbb{R}^n$ when $n=0$?

Question

My teacher ask this question, though I thought $\mathbb R^n$ is Euclidean space with $n\geq 1$ but I didn't find this $\mathbb R^0$ space in web search. Can anybody help me out?

Thanks in advance.

Answer 1

The notation $\Bbb{R}^n$ is indicative of the notation $X^Y$ for the set of all functions $f : Y \to X$ in set theory. Specifically, every $n$-tuple of real numbers can be seen as a mapping from $[n] := \{ 1,2,\dots,n \}$ to $\Bbb{R}$, and conversely, every such function defines an $n$-tuple of real numbers. So, $\Bbb{R}^n$ can be identified with $\Bbb{R}^{[n]}$.

Hence, it makes sense to identify $\Bbb{R}^0$ with $\Bbb{R}^{\emptyset}$, the set of all functions from the empty set (the set with $0$ elements) to $\Bbb{R}$. Since, there is only one function with domain as the empty set, namely the empty function, $\Bbb{R}^0$ has a single element.

Thus, one can take $\Bbb{R}^0$ to be a set with a single element. Since there is only one vector space with a single element, namely the zero vector space, it is reasonable to identify $\Bbb{R}^0$ with the zero vector space $\{ 0 \}$.

Answer 2

Recall that the vector space $\mathbb R^n$ is the set of all points $(a_1, ... , a_n), \space a_i \in \mathbb R \space \forall i \in I_n$ where $I_n$ is the indexing set to the nth element, that satisfy the vector space axioms.

Also note that, a vector space can be factored through the cartesian cross product as follows: $\mathbb R^{n + m} = \mathbb R^n \times \mathbb R^m$.

Thus, we can denote any space $\mathbb R^n$ as $\mathbb R^n = \mathbb R^n \times \mathbb R^0$. So, we can think of the vector space $\mathbb R^0$ as an identity under the cross product. So we can think of $\mathbb R^0$ as the trivial vector space, with only one object: the zero vector.

Note that any space $\mathbb R^n$ is the set of all points $(a_1, ... , a_n, 0, 0, ...)$ in a higher-dimensional space $\mathbb R^m \supset \mathbb R^n$ (think about how a 2-dimensional space is simply a 3-dimensional space with the z coördinate set to zero). Clearly, the space $\mathbb R^1$ is the set of all points $(a_1)$ ($(a_1, 0, 0, ..., 0_n) \in \mathbb R^n$).

Therefore, the space $\mathbb R^0$ is the trivial/zero space, in that it only contains the zero vector.