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)

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) 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" (denoted with an uppercase ) is reserved for application with a function ,

(4)

with denoting an angle bracket.