# Intuition behind $F^n$ and $F^\infty$ being examples of function spaces, $F^S$.

I'm working through Axler's linear algebra textbook and in it he says that $$F^n$$, the set of n-tuples with elements in $$\mathbb{R}$$ or $$\mathbb{C}$$, as well as $$F^\infty$$, are specific cases of $$F^S$$ - the set of functions mapping from $$S$$ to $$F$$. He states that we can think of $$F^n$$ as $$F^{\{1,2,...,n\}}$$ and analogously for $$F^\infty$$.

I am having trouble understanding how the set of all points (vectors) in n-dimensions (i.e., $$F^n$$) is equivalent to the set of all functions mapping the first $$n$$ natural numbers to $$F$$.

Consider the bijection $$(\eta_1, \ldots, \eta_n) \mapsto (k \mapsto \eta_k)$$ where $$k$$ runs through $$1, \ldots, n.$$ In other words, given a vector $$\eta = (\eta_1, \ldots, \eta_n) \in \mathbf{K}^n$$ consider the function $$\varphi_\eta:\{1, \ldots, n\} \to \mathbf{K}$$ given by $$\varphi_\eta(k)=\eta_k.$$ The bijection is $$\eta \mapsto \varphi_\eta;$$ the inverse is given by $$\psi \mapsto (\psi(1), \ldots, \psi(k)).$$
• I think I understand now. Is there an explanation related to cardinality as to why the set of size-$n$ vectors form a one-to-one correspondence with the set of functions {$f:\{1,2,..,n\} \to$ K} ? Feb 27, 2019 at 19:26
• Uh, the number of functions $f:\{1, \ldots, n\} \to \mathbf{K}$ is infinity (consider constant functions. Feb 27, 2019 at 20:05