Lp norm ($\Vert \cdot {\Vert }_p$)

For a measurable function $f$, its $L^p$ norm iswhen $1\le p<\infty$, while $\Vert f{\Vert }_{\infty }$ is its essential supremum.

The displayed quantity is homogeneous and separates zero, but generally violates the triangle inequality. Its $q$th power is subadditive because $(a+b{)}^q\le a^q+b^q$ for nonnegative $a,b$. Minimizing it is equivalent to minimizing its $q$th power. On finite vectors it is a standard nonconvex sparsity penalty.

For $1\le p\le \infty$, the Lp norm satisfies the triangle inequality. For $1<p<\infty$, apply the Holder inequality to $\int (|f|+|g|)|f+g{|}^{p-1}$, then divide by $\Vert f+g{\Vert }_p^{p-1}$ when it is nonzero. The cases $p=1$ and $p=\infty$ follow directly from the pointwise triangle inequality. In particular, a series whose term norms have a finite sum converges in the complete $L^p$ space.

The $L^2$ norm isIt is induced by the $L^2$ inner product and is the continuous analogue of the Euclidean norm.

For independent mean-zero random variables $X_1,\dots ,X_n$ and $p\ge 1$, the Marcinkiewicz–Zygmund inequality compares the $L^p$ norm of their sum with the square function:For $p=2m$, one may take $C_p^{1/p}=O(\sqrt{m})$.

For conjugate exponents $p,q\in [1,\infty ]$, Hölder's inequality states $\Vert fg{\Vert }_1\le \Vert f{\Vert }_p\Vert g{\Vert }_q$. For finite vectors, the case $p=1$, $q=\infty$ is $|x^{T}y|\le \Vert x{\Vert }_1\Vert y{\Vert }_{\infty }$.