VNorms - Norms of Vectors When we want to measure the...

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Norms of Vectors When we want to measure the length of, or distance between, vectors we need a yardstick that measures in a consistent way, generalizing the idea of absolute value to vector spaces. We can capture the essence of length by requiring that such a yardstick satisfy the following properties: 1. f ( x ) > 0 , x 6 = 0, 2. f ( αx ) = | α | f ( x ) , x and for all scalars α , and 3. f ( x + y ) f ( x ) + f ( y ) , x and y . Any function which satisfies these properties can be called a norm , and we usually use the symbol f ( x ) = k x k . With a norm, R n is a metric space, with the distance between x and y being d ( x, y ) = k x - y k . The unit circle or unit ball in ( R n , k · k ) is the set of vectors with unit length: { x R n : k x k = 1 } . A norm is completely determined by its unit circle (property 2), so don’t be suprised if almost all unit circles aren’t. .. circular. You have already seen the Euclidean norm, it is also called the 2-norm because it is the p = 2 member of this family of p-norms:
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