Kunze Linear Algebra · Problem 8.2.4

Inner Product Spaces

Mathematics — Methods · inner-product-spaces-distance

Problem

Let VV be an inner product space. The distance between two vectors α\alpha and β\beta in VV is defined by d(α,β)=∥α−β∥.d(\alpha, \beta) = \|\alpha - \beta\|. Show that (a) d(α,β)≥0d(\alpha, \beta) \geq 0; (b) d(α,β)=0d(\alpha, \beta) = 0 if and only if α=β\alpha = \beta; (c) d(α,β)=d(β,α)d(\alpha, \beta) = d(\beta, \alpha); (d) d(α,β)≤d(α,γ)+d(γ,β)d(\alpha, \beta) \leq d(\alpha, \gamma) + d(\gamma, \beta).

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