[[Function space]]
# Function algebra

Let $\mathbb{K}$ be a [[field]] and $X$ be some set.
The **function algebra** $\mathbb{K}^X = \Set(X,\mathbb{K})$ consists of functions $X \to \mathbb{K}$ with scaling, addition, and multiplication defined pointwise. #m/def/falg

## Additional structure

- If $X$ is a topological space, one may consider a [[Continuous function algebra]].
- If one restricts to maps of finite [[Support of a map|support]] one gets the [[free module]] of $X$ over $\mathbb{K}$.
  - In particular, if $X$ is a group this is the [[group ring]].

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