---
title: "F^s is a Vector Space Over F"
type: concept
source: https://www.jemoka.com/posts/kbhfs_is_a_vector_space/
confidence: high
status: active
---

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}
closeness of scalar multiplication \begin{equation} (\lambda f)(x)=\lambda f(x), \forall \lambda \in \mathbb{F}, f \in \mathbb{F}^{S}, x \in S \end{equation}
commutativity inherits \(\mathbb{F}\) (for the codomain of functions \(f\) and \(g\))
associativity inherits \(\mathbb{F}\) for codomain or is just \(\mathbb{F}\) for scalar
distribution inherits distribution in \(\mathbb{F}\) on the codomain again
additive identity \begin{equation} 0(x) = 0 \end{equation}
additive inverse \begin{equation} (-f)(x) = -f(x) \end{equation}
multiplicative identity \(1\) hee hee
