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Of particular importance are the norms

  1. The 1-norm (or “taxi-cab norm”): \(\|\vec x\|_1 = \sum_{i=1}^n |x_i|\text{.}\)
  2. The \(2\)-norm (a.k.a. the Euclidean norm): \(\|x\|_2 = \bigg(\sum_{i=1}^n |x_i|^2\bigg)^{1/2}.\) Note, if \(x \in \Cm \text{,}\) then \(x^\star x = \|x\|_2^2 \text{.}\)
  3. The \(\infty\)-norm (also known as the max-norm): \(\|\vec x\|_\infty = \max_{i=1}^n |x_i|\text{.}\)
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