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:

${\displaystyle \rho (x,y)_{L_{2}}=\max |f(x)-g(x)|}$

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

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

Cp Space

The Cp is the set of all continuous functions for which the first p derivatives are also continuous. If ${\displaystyle p=\infty }$  the function is called "infinitely continuous. The set ${\displaystyle C^{\infty }}$  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:

${\displaystyle \int _{a}^{b}|f(x)|dx<\infty }$

L p Space

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

${\displaystyle \int _{a}^{b}|f(x)|^{p}dx<\infty }$

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