#scalarproductnorm #scalarproduct #triangleinequality #cauchyschwarz
Unit 1 · 8.1 · Vectors, norms & sequences
Scalar product & Euclidean norm The scalar product combines two vectors into a number; the Euclidean norm measures length. The triangle and Cauchy–Schwarz inequalities are the two estimates everything later relies on.
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Formulas at a glance · Vectors, norms & sequences u ⋅ v = ∑ i = 1 n u i v i , ‖ v ‖ 2 = v ⋅ v , | u ⋅ v | ≤ ‖ u ‖ 2 ‖ v ‖ 2
‖ v ‖ ∞ ≤ ‖ v ‖ 2 ≤ d ‖ v ‖ ∞ , 1 d ‖ v ‖ 1 ≤ ‖ v ‖ 2 ≤ ‖ v ‖ 1