1 Norm Review
A norm is a function defined on a vector space that satisfies several properties including non-negativity and the triangle inequality. Formally, a norm is defined as follows.
For vector space \(V\) on field \(\mathbb{R}\) a norm is a function \(||\cdot||:V\rightarrow\mathbb{R}\) satisfying the following properties for \(v,x\in V\): 1. \(\Vert v\Vert \geq 0~\forall v\in V\) 2. \(||v||=0\) iff \(v=0\) 3. \(\Vert\alpha v \Vert =|\alpha|\Vert v\Vert\) for \(\alpha\in\mathbb{R}\) 4. \(\Vert v+z\Vert\leq \Vert v\Vert + \Vert x \Vert\) (triangle inequality)
A norm is a very general construction and \(p\)-Norms are a specific type of norm generalizing the concept of the [[euclidean-norm]]. ## p Norm in Finite Dimensional Vector Space
The concept of a norm arose from the generalization of the Euclidean length of a vector (i.e. distance from the origin) in the \(n\) dimensional real vector space \(\mathbb{R}^n\) which is given by the [[euclidean-norm]] \[ \|x\|_{2}=(x_{1}^2+x_{2}^2+\dots +x_{n}^2)^{1/2}. \] The Euclidean distance between two points \(x\) and \(y\) is the length \(\|x-y\|_{2}\) of the straight line between the two points.
Given \(n\)-dimensional vector space \(V\) and real number \(p\geq 1\), the \(p\)-norm (or \(L^p\)-norm) of \(x\in V\) is the norm specified by \[||x||_{p}=(|x_{1}|^p + \dots +|x_{n}|^p)^{1/p}.\]
- For \(p=1\) we obtain the taxicab norm.
- For \(p=2\) we obtain the [[euclidean-norm]].
- For \(p\rightarrow\infty\) the \(p\)-norm approaches the infinity norm (maximum norm).
Note that the absolute value bars can be dropped when \(p\) is a rational number with an even numerator in its reduced form.
The \(L_{\infty}\)-norm or maximum norm is the limit of the \(L_{p}\)-norms for \(p \to \infty\) and is given by \[\|x\|_{\infty}= \max\left\{ |x_{1}|,\dots ,|x_{n}| \right\}.\]
2 Lp Norm in Function Space
Function spaces can be though of as infinite dimensional vector spaces and as such, we are able to define norms on them.
Given a probability space \((X,\mathcal{F}, \mu)\) and a real number \(p \in [1, \infty)\), the \(L^p\)-norm of a measurable function \(f:X\to \mathbb{R}\) is defined by \[\|f\|_{p}=\left( \int_{X}|f|^p~d\mu \right)^{1/p}. \]
Note that the \(L^p\)-norm of a function \(f\) may be either finite or infinite. The \(L^p\) functions are those for which the \(p\)-norm is finite.
Let \((X, \mathcal{F}, \mu)\) be a measure space, and let \(p \in [1,\infty)\). An \(L^p\) function on \(X\) is a measurable function \(f\) on \(X\) for which \[\int_{X}|f|^p ~d\mu<\infty.\]
- For \(p\rightarrow\infty\) we obtain the supremum norm \(\Vert f\Vert_{\infty}= \sup\{|f(x)|: x\in X\}\)
- From this result we see the relationship between functions as infinite dimensional vectors.
- This norm is integral in probability theory as it defines Lp Spaces.
The \(L_{\infty}\)-norm or supremum norm is the limit of the \(L_{p}\)-norms for \(p \to \infty\) and is given by \[\lVert f \rVert_{\infty} := \sup\{|f(x)|: x \in X\}. \]