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

Created over 8 years ago, updated 10 days ago

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.