L2normi

L2normi, often written as L2-normi in Finnish literature, refers to the L2 norm, a standard measure of magnitude or length in mathematics. It is commonly used for vectors in Euclidean space and, more generally, for functions in L2 spaces. The L2 norm is also called the Euclidean norm in many contexts.

For a vector x = (x1, x2, ..., xn) in R^n, the L2normi is defined as the square root of the sum of the squares of its components: ||x||2 = sqrt(x1^2 + x2^2 + ... + xn^2). This norm arises from the inner product <x, x> = sum xi^2, since ||x||2 = sqrt(<x, x>). It satisfies non-negativity, is zero only for the zero vector, is homogeneous, and obeys the triangle inequality. For example, ||(3, 4)||2 = 5.

In functional analysis, the L2 norm extends to functions in L2[a,b], where ||f||2 = sqrt(∫a^b f(t)^2 dt). The inner product is defined by <f, g> = ∫a^b f(t)g(t) dt, and ||f||2 = sqrt(<f, f>). L2 norms induce the L2 distance between objects, which is fundamental in approximation, signal processing, and statistics.

Applications of the L2normi include measuring vector lengths and distances, regularization in machine learning (L2 or ridge regularization), projection onto subspaces, and assessing approximation errors. The norm is sensitive to large components, which is a consideration when portability to outlier-prone data is important.