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Engineering Analysis/Function Spaces

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Function SpaceEdit

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.

ContinuityEdit

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 SpacesEdit

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

C SpaceEdit

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

The metric for C space is defined as:

 

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

 

Cp SpaceEdit

The Cp is the set of all continuous functions for which the first p derivatives are also continuous. If   the function is called "infinitely continuous. The set   is the set of all such functions. Some examples of functions that are infinitely continuous are exponentials, sinusoids, and polynomials.

L SpaceEdit

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:

 

L p SpaceEdit

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:

 

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