Engineering Analysis/Function Spaces

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Function Space[edit]

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[edit]

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[edit]

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

C Space[edit]

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[edit]

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[edit]

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[edit]

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".