I was reading the John Watrous book. In the second page, he mentions that an example of an alphabet would be $\Sigma = \{0, 1\}$ and a complex Euclidean space $C^\Sigma$ would be the set of all functions that maps an element from $\Sigma$ to complex numbers $C$. Could anyone give an example of such a function? That is, an element from $C^\Sigma$?
Also, when the alphabet is defined as $\Sigma = \{1,...n\}$, let's say for n = 2, it's $\Sigma = \{1,2\}$. Then what would the $C^\Sigma$ look like? Any example?
Thanks.
There is no bigger hidden meaning in ${\mathbb C}^\Sigma$ in that it is just the set of functions from $\Sigma$ to $\mathbb C$.
If you are searching for a way to represent a function $f:\Sigma_n\to\mathbb C$, where $\Sigma_n=\{1,\dots,n\}$, you may put it as a table of its values
\begin{array}{c|c} \Sigma &\mathbb C\\ \hline 1& f(1)\\ \hline 2& f(2)\\ \hline \vdots& \vdots\\ \hline n & f(n) \end{array}
Another possibility, using the particular structure of $\Sigma_n$ would be to just represent such functions as complex vectors $(c_1,\dots,c_n)$, where this vector then represents the function $f$ where $f(i)=c_i$.
Technically (from a set-theoretic perspective) every function is just a relation between is domain and range. To be more precise, we "encode" a function $f:X\to Y$ for sets $X,Y$ as the relation $F\subseteq X\times Y$ where
$$F=\{(x,f(x))\mid x\in X\}$$
From this perspective, ${\mathbb C}^{\Sigma_n}$ is just the set of all such relations, i.e. it is a set containing sets which are relations encoding functions.
Using the encoding of functions from $\Sigma_n$ to $\mathbb C$ as vectors $(c_1,\dots,c_n)$, as every function can be represented as such a vector and every vector represents such a function, you can think of ${\mathbb C}^{\Sigma_n}$ as the set of vectors $\mathbb C^n$ since, as just described, there is a bijection between them. Note however that from a set theoretic perspective, ${\mathbb C}^{\Sigma_n}$ really is the set described above and not $\mathbb C^n$. If you are however not interested in the actual set-theoretic structure of ${\mathbb C}^{\Sigma_n}$, then I think $\mathbb C^n$ is more customary.