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Functional spaces. Examples.

Def. Function space. A function space is a family of functions having a common domain of definition. The family is viewed as a space and the individual functions as points in the space.

Syn. Functional space

Def. Linear function space. A function space is linear if it meets the axiomatic requirements of a linear space. See linear space.

Theorem. A function space is a linear space if and only if whenever *f*_{1}, * f*_{2}, .... , *f*_{n} are members
of S and α_{1}, α_{2}, ..... , α_{n} are arbitrary real numbers, the linear combination α_{1} *f*_{1} + α_{2} *f*_{2} + .... + α_{n} *f*_{n}
is also a member of S.

Examples of function spaces that are linear. Each of the following sets of functions, assumed to be single-valued, real-valued and defined over a common closed interval I = [a, b] is a linear space.

1. set of all real-valued functions

2. set of all bounded functions

3. set of all Riemann integrable functions

4. set of all sectionally continuous functions

5. set of all continuous functions

6. set of all functions with a continuous n-th derivative, where n is a positive integer

7. set of all functions with derivatives of all orders

8. set of all polynomials

9. set of all constant-valued functions

● Every space in the above list contains all spaces that follow it and thus is contained in all spaces that precede it.

Example of a function space that is not linear: the set of monotonic functions on a given closed interval. For example, the two functions sin x - 2x and 2x are separately monotonic on the interval [0, 2π], but their sum is not.

Functions in a function space can be viewed as points in an infinite-dimensional space. A function space by itself, however, contains no concept of distance between points of the space. Usually we want a way to define what we mean by the proximity of two functions i.e. the distance between two points of the infinite-dimensional space. We accomplish this by defining a distance function on the space. We thus turn the space into a metric space. Two distance functions that are often used are

and

Others can be used. Thus distinct methods of defining the distance leads to distinct infinite-dimensional spaces.

References

Olmstead. Real Variables. pp. 538-539

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