Function Spaces
Function Space (S → T)
The function space S → T is the set of all functions f such that dom(f) ⊆ S and ran(f) ⊆ T.
If f ∈ S → T, then f is a function from S to T.
Example: f ∈ N → N means f maps natural numbers to natural numbers.
Key Idea
A function space describes all possible functions between two sets.
Example: Function f
From earlier examples:
f ∈ N → N
f ∈ {0,1,2,3,4} → N
f ∈ N → {2,3,4,5,6}
Example: Function g
g ∈ N → N
g ∈ N → {0,1,2,3,4}
Important Insight
Every function can belong to multiple function spaces depending on allowed inputs and outputs.