Concept guide
Inner-product-induced norm: proving √⟨x,x⟩ is a norm and computing exactly for a positive-definite matrix on ℝ²
In any real inner-product space, prove that N(v)=√⟨v,v⟩ is a norm: nonnegativity and separation, absolute homogeneity, and the triangle inequality (via a fully proved Cauchy–Schwarz inequality). Then, on ℝ² with the inner product given by A=[[2,1],[1,2]], compute exactly the squared norms and the squared distance for x=(1,0) and y=(0,1), and check Cauchy–Schwarz, the triangle inequality and the parallelogram identity on this instance.