Prerequisites

Random Variables, Expected Value

Summary

A function space is merely a set of functions, or a set of equivalence classes of functions. This document will introduce the important $L^p$ function spaces and state their most basic properties.

Function Spaces

Banach spaces: Let $X$ be a normed linear space with norm $\vert\vert\cdot\vert\vert$. If $X$ is complete ¹ with respect to the induced metric $d(x,y) := \vert\vert x-y\vert\vert$, then the pair $(X, \vert\vert\cdot \vert\vert)$ is called a Banach space.

Hilbert spaces: Let $K$ be a linear space with an inner product $\langle \cdot,\cdot \rangle$. If $K$ is complete with respect to the induced metric $d(x,y) := \sqrt{\langle x-y,x-y\rangle}$, then $(K, \langle \cdot, \cdot \rangle)$, is called a Hilbert space.

There are plenty of Banach spaces and Hilbert spaces that are not function spaces. For example, ${\mathbb{R}}^n$ equipped with the usual inner product is a Hilbert space, and hence a Banach space. Points of ${\mathbb{R}}^n$ (considered simply as points) are not functions, so ${\mathbb{R}}^n$ is not a function space. However, there is a large class of function spaces that are either Hilbert spaces or Banach spaces.

$L^p(\mu,S)$ spaces: Let $(S, \mathcal{S}, \mu)$ be a measure space. Set

$$\mathcal{L}^p(\mu,S) := \{f:S\to {\mathbb{R}} f~ \mu , \int \vert f\vert^p d \mu<\infty \}.$$

When the measureable space and measure can be identified by from the context of the discussion, $\mathcal{L}^p(S, \mathcal{S}, \mu)$ is usually abreviated to $\mathcal{L}^p$. One usually places the following equivalence relation on $\mathcal{L}^p$: write $f \sim g$ iff $f, g \in \mathcal{L}^p$ and

$$\int \vert f-g\vert^p d\mu = 0.$$

This equivalence relation partitions $\mathcal{L}^p$ into equivalence classes of functions, and functions belonging to the same equivalence class are equal $\mu$-almost everywhere. Thus we have a new space, $L^p(S, \mathcal{S}, \mu)$ consisting of equivalence classes of functions in $\mathcal{L}^p$, and we write $[f] \in L^p$ to express the assertion that an equivalence class $[f]$ is in this set. Such precision is rarely needed, however, and authors usually write $f$ instead of $[f]$.²

Note that $L^p$ is a linear space because because sums of integrable functions are integrable, and $cf$ is integrable if $f$ is integrable and $c$ is a constant. Let $\vert\vert[X] \vert\vert _p := (\int \vert X\vert^p)^{1/p}$ be the $L^p$ norm of $X$; this is an honest-to-goodness norm ³. Define convergence in $L^p$ as follows:

$$X_n \stackrel{L^p}{\rightarrow} X \textrm{ means } \vert\vert X_n - X\vert\vert _p \rightarrow 0$$ (1)

It can be shown that $L^p$ is complete. Therefore $L^p$ spaces are Banach spaces for $p \geq 1$.

$L^2$ is a Hilbert space if we give it the inner product $\langle f,g\rangle = \int fg\ d\mu$. You can check that the norm induced by this inner product agrees with our previous definition of the $L^2$ norm.

For $p=1$ $L^p$ corresponds with the space of integrable functions

$$L^1(\Omega, {\cal F},{\mathbb{P}}) := \{X : X \ {\mathbb{E}}(\vert X\vert)<\infty\}.$$ (2)

References

Real and Complex Analysis, Walter Rudin
Wikipedia: $L^p$ spaces. http://en.wikipedia.org/wiki/Lp_space

Footnotes

...http://planetmath.org/?op=getobj&from=objects&id=3193 ¹

Recall that a metric space is complete if every Cauchy sequence converges to a limit.

....²

You do something similar every time you write $\frac{1}{2}$ instead of $[\frac{1}{2}]$, because everyone knows that a rational number $\frac{p}{q}$ is an equivalence class of ordered pairs. $\frac{1}{2} = \frac{2}{4} = \frac{3}{6} = \cdots$ etc.

... norm³

This definition is independent of the choice of representative of the equivalence class, i.e. if $[Y] = [X]$ then we could replace $X$ by $Y$ in the integral above and the result would be the same.