Let $B(X)$ be the set of bounded real valued functions on the set X. And let X be the finite set $\{1,...,n\}$.
I have to find a bijection between $B(X)$ and $\mathbb R^{n}$, however I'm struggling to understand say, what is $B(1)$? For example.
The way I'm trying to think of it is, I am trying to find a function $\phi$, which takes $n$ bounded real function to the reals. $$\phi: B(X) \rightarrow \mathbb R^{n}, f \rightarrow \phi(f) = \{a_1,a_2,...,a_n\}$$ with $f = \{f_1,f_2,...,f_n\}\in B(X)$ and $a_i\in \mathbb R$.