The set of all functions on a set is a vector space.
Created almost 9 years ago, updated 5 months ago
ID24100
Theorem: Let (X) be any set and (\mathbb{F}) a field. The set of all functions (f: X o \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.
**Theorem:** Let \(X\) be any set and \(\mathbb{F}\) a field. The set of all functions \(f: X o \mathbb{F}\), 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.