Concept guide
Real normed vector space: universal norm certificates for the ℓ¹ and ℓ^∞ formulas, and two formulas that are not norms
On ℝⁿ (n≥1 fixed), for all vectors u, v and all reals a, prove that ‖v‖₁=Σ|v_i| and ‖v‖_∞=max|v_i| both satisfy the three norm axioms: nonnegativity with ‖v‖=0 ⇔ v=0, ‖av‖=|a|‖v‖, and ‖u+v‖≤‖u‖+‖v‖. Then, in ℝ³, compute both norms and instances of the triangle inequality and homogeneity for the integer-coordinate vectors u=(3,−4,1) and v=(−1,2,5); finally show that φ(v)=|v₁| and ψ(v)=(‖v‖₁)² are not norms.