## Norms

Short Summary: Norm refers to the magnitude of a vector. It commutes with scaling, obeys a form of the triangle inequality, and is zero only at the origin. Since the definition of the magnitude can vary, there are many different norms.

### L0 Norm

The L0 Norm is the number of non-zero elements in a vector. L0 Norm is not a true norm. It satisfies only two of the three necessary properties of a norm: that it is zero only for the all-zero vector, and the triangle inequality. It is not, however, positively homogeneous. And unlike all true norms, it is not convex. Calling it a "norm" sews confusion. L0 Norm is the cardinality function. The L0 norm is popular in compressive sensing which tries to find the sparsest solution to an underdetermined set of equations:

\begin{equation} \lVert X \rVert_0 = \lvert\{x \in X \mid x \neq 0\}\rvert \end{equation}

### L1 Norm

The L1 Norm, also known as Manhattan Distance or Taxicab norm, is the sum of the magnitudes of the vectors in a space:

\begin{equation} \lVert X \rVert_1 = \sum_{x \in X} \text{abs}(x) \end{equation}

### L2 Norm

The L2 Norm, also known as Euclidean norm, represents the shortest distance to the center:

\begin{equation} \lVert X \rVert_2 = \sqrt{\sum_{x \in X} x^2} \end{equation}

### L-Infinity Norm

The L-Infinity Norm is the element of the greatest magnitude:

\begin{equation} \lVert X \rVert_\infty = \max(\text{abs}(X)) \end{equation}