street plan who should measure distance not in terms of the length of the straight line to their destination, but in terms of the rectilinear distance, which takes into account that streets are either orthogonal or parallel to each other. The class of -norms generalizes these two examples and has an abundance of applications in many parts of mathematics, physics, and computer science. Definition For a real number the -norm or -norm of is defined by The absolute value bars can be dropped when is a rational number with an even numerator in its reduced form, and is drawn from the set of real numbers, or one of its subsets. The Euclidean norm from above falls into this class and is the -norm, and the -norm is the norm that corresponds to the rectilinear distance. The -norm or maximum norm (or uniform norm) is the limit of the -norms for It turns out that this limit is equivalent to the following definition: See -infinity. For all the -norms and maximum norm as defined above indeed satisfy the properties of a "length function" (or norm), which are that: only the zero vector has zero length, the length of the vector is positive homogeneous with respect to multiplication by a scalar (positive homogeneity), and the length of the sum of two vectors is no larger than the sum of lengths of the vectors (triangle inequality). Abstractly speaking, this means that together with the -norm is a normed vector space. Moreover, it turns out that this space is complete, thus making it a Banach space. This Banach space is the -space over Relations between -norms The grid distance or rectilinear distance (sometimes called the "Manhattan distance") between two points is never shorter than the length of the line segment between them (the Euclidean or "as the crow flies" distance). Formally, this means that the Euclidean norm of any vector is bounded by its 1-norm: This fact generalizes to -norms in that the -norm of any given vector does not grow with : For the opposite direction, the following relation between the -norm and the -norm is known: This inequality depends on the dimension of the underlying vector space and follows directly from the Cauchy–Schwarz inequality. In general, for vectors in where This is a consequence of Hölder's inequality. When In for the formula defines an absolutely homogeneous function for however, the resulting function