# Concept derivation guides

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Concept guide



## [Inner-product-induced norm: proving √⟨x,x⟩ is a norm and computing exactly for a positive-definite matrix on ℝ²](/en/derivations/derivation-mko-inner-product-induced-norm-01/?from=%2Fen%2Fderivations%2F%3Fq%3Dmko-inner-product-induced-norm)

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.

Inner-product-induced norm

derivation-mko-inner-product-induced-norm-01





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