3. Normed Spaces

Question 1

Define what it means for ‖ · ‖ to be a norm on X.

Answer 1

Definition 3.1:
A norm is a map ‖ · ‖ : X → R such that
* ‖x‖ > 0 for all x ∈ X \ {0};
* ‖λx‖ = |λ|‖x‖ for all x ∈ X, λ ∈ F;
* ‖x + y‖ ≤ ‖x‖ + ‖y‖ for all x, y ∈ X.