The set of all functions on a set is a vector space.

**Theorem:** Let \(X\) be any set and \(\mathbb{F}\) a field. The set of all functions \(f: X \to \mathbb{F}\), denoted \(\mathbb{F}^X\), is a vector space over \(\mathbb{F}\) with pointwise operations:

- **Addition:** \((f + g)(x) = f(x) + g(x)\) for all \(x \in X\)
- **Scalar multiplication:** \((cf)(x) = c \cdot f(x)\) for all \(x \in X\)

**Zero vector:** The zero function \(z(x) = 0\) for all \(x\).
**Additive inverse:** \((-f)(x) = -f(x)\).

**Verification:** All axioms follow from the field properties of \(\mathbb{F}\) applied pointwise.

**Example:** When \(X = [0,1]\), this gives the space of all real-valued functions on the unit interval, which includes continuous functions, polynomials, and many more.Copy