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 &epsilon; > 0 there exists a &delta;(&epsilon;) > 0 such that |f(x) - f(x0)| < &epsilon; when |x - x0| < &delta;(&epsilon;).

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):


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

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


 * $$\int_a^b |f(x)|dx < \infty$$

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:


 * $$\int_a^b |f(x)|^pdx < \infty$$

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