F^s is a Vector Space Over F

We define a set \(\mathbb{F}^{s}\), which is the set of unit functions that maps from any set \(S\) to \(\mathbb{F}\).

closeness of addition

\begin{equation} (f+g)(x) = f(x)+g(x), \forall f,g \in \mathbb{F}^{S}, x \in S \end{equation}

\begin{equation} (\lambda f)(x)=\lambda f(x), \forall \lambda \in \mathbb{F}, f \in \mathbb{F}^{S}, x \in S \end{equation}

inherits \(\mathbb{F}\) (for the codomain of functions \(f\) and \(g\))

inherits \(\mathbb{F}\) for codomain or is just \(\mathbb{F}\) for scalar

inherits distribution in \(\mathbb{F}\) on the codomain again

\begin{equation} 0(x) = 0 \end{equation}

\begin{equation} (-f)(x) = -f(x) \end{equation}

\(1\) hee hee