L^2-Norm

The -norm (also written "-norm") is a vector norm defined for a complex vector

(1)

by

(2)

where on the right denotes the complex modulus. The -norm is the vector norm that is commonly encountered in vector algebra and vector operations (such as the dot product), where it is commonly denoted . However, if desired, a more explicit (but more cumbersome) notation can be used to emphasize the distinction between the vector norm and complex modulus together with the fact that the -norm is just one of several possible types of norms.

For real vectors, the absolute value sign indicating that a complex modulus is being taken on the right of equation (2) may be dropped. So, for example, the -norm of the vector is given by

(3)

For real vectors, the set is a disk of radius 1 in two dimensions and a unit ball in three dimensions. Its boundary is a unit circle or unit sphere, respectively.

The -norm is also known as the Euclidean norm. However, this terminology is not recommended since it may cause confusion with the Frobenius norm (a matrix norm), which is also sometimes called the Euclidean norm. The -norm of a vector is implemented in the Wolfram Language as Norm[m, 2], or more simply as Norm[m].

The -norm of a square integrable measurable function on a measure space is

(4)

where is the Hermitian inner product, written using angle brackets. Functions equal almost everywhere are identified as one equivalence class in the L²-space, so a function of norm zero represents the zero class (Hunter 2011, p. 79).

With counting measure on a finite set, this formula gives the finite-vector -norm. With counting measure on the positive integers, it gives the -norm of a sequence whose squared complex moduli have a finite sum. Thus, the uppercase and lowercase conventions describe related function and sequence settings, rather than different notions (Hunter 2011, p. 79).