Engineering Analysis/Function Spaces

Function Space

A function space is a linear space where all the elements of the space are functions. A function space that has a norm operation is known as a normed function space. The spaces we consider will all be normed.

Continuity

f(x) is continuous at x₀ if, for every ε > 0 there exists a δ(ε) > 0 such that |f(x) - f(x₀)| < ε when |x - x₀| < δ(ε).

Common Function Spaces

Here is a listing of some common function spaces. This is not an exhaustive list.

C Space

The C function space is the set of all functions that are continuous.

The metric for C space is defined as:

$\rho (x,y{)}_{L_2}=\max |f(x)-g(x)|$

Consider the metric of sin(x) and cos(x):

$\rho (sin(x),cos(x){)}_{L_2}=\sqrt{2},x=\frac{3\pi }{4}$

C^p Space

The C^p is the set of all continuous functions for which the first p derivatives are also continuous. If $p=\mathrm{\infty }$ the function is called "infinitely continuous. The set $C^{\mathrm{\infty }}$ is the set of all such functions. Some examples of functions that are infinitely continuous are exponentials, sinusoids, and polynomials.

L Space

The L space is the set of all functions that are finitely integrable over a given interval [a, b].

f(x) is in L(a, b) if:

$\underset{a}{\overset{b}{\int }}|f(x)|dx<\mathrm{\infty }$

L _p Space

The L_p space is the set of all functions that are finitely integrable over a given interval [a, b] when raised to the power p:

$\underset{a}{\overset{b}{\int }}|f(x){|}^{p}dx<\mathrm{\infty }$

Most importantly for engineering is the L₂ space, or the set of functions that are "square integrable".