Concept guide
Real Banach space: norm, induced metric and completeness of (ℝ², ℓ¹)
On V=ℝ² take the ℓ¹ norm ‖x‖₁=|x₁|+|x₂| and the metric induced by that same norm, d₁(x,y)=‖x−y‖₁. (1) Prove that ‖·‖₁ is a norm and d₁ a metric, and compute d₁((3,−1),(1,2)). (2) Prove completely, from the Cauchy completeness of the reals, that (ℝ²,d₁) is complete, so (ℝ²,‖·‖₁) is a real Banach space; also cite the named theorem that finite-dimensional real normed spaces are complete. (3) Prove that ‖·‖₁ violates the parallelogram law and hence is induced by no inner product.