# 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):

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

### 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".