Maximumnorm

Maximumnorm, more commonly known as the infinity norm or L-infinity norm, is a way of measuring the size of a vector or the maximal effect of a matrix on a vector. For a vector x in R^n, it is defined as ||x||∞ = max_i |x_i|, the largest absolute value among its components. For a matrix A, the corresponding operator norm induced by the vector infinity norm is ||A||∞ = max_i ∑_j |a_ij|, the maximum absolute row sum.

The infinity norm is the limit of the p-norms ||x||p = (∑|x_i|^p)^{1/p} as p approaches infinity. It is also referred to as the Chebyshev or uniform norm in some contexts. As a norm, it satisfies the standard properties: non-negativity, definiteness, homogeneity, and the triangle inequality.

For vectors, the maximumnorm bounds each component: |x_i| ≤ ||x||∞ for all i. It is related to other norms by standard finite-dimensional equivalences, such as ||x||2 ≤ √n ||x||∞ and ||x||∞ ≤ ||x||2. These relationships are useful in analysis and numerical estimates.

In matrix analysis, the induced infinity norm ||A||∞ provides a measure of how A can amplify the infinity norm of a vector, and it equals the maximum row-wise sum of absolute values. This makes it convenient for bounding errors and assessing stability in iterative methods, as well as in optimization problems that penalize worst-case deviations.

Computationally, evaluating ||x||∞ requires a single pass through the vector to find the maximum absolute component, which is efficient and straightforward.