SolitaryRoad.com

Website owner:  James Miller


[ Home ] [ Up ] [ Info ] [ Mail ]

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 f1, f2, .... , fn are members of S and α1, α2, ..... , αn are arbitrary real numbers, the linear combination α1 f1 + α2 f2 + .... + αn fn 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


             ole.gif


and



             ole1.gif



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




References

  Olmstead. Real Variables. pp. 538-539



More from SolitaryRoad.com:

The Way of Truth and Life

God's message to the world

Jesus Christ and His Teachings

Words of Wisdom

Way of enlightenment, wisdom, and understanding

Way of true Christianity

America, a corrupt, depraved, shameless country

On integrity and the lack of it

The test of a person's Christianity is what he is

Who will go to heaven?

The superior person

On faith and works

Ninety five percent of the problems that most people have come from personal foolishness

Liberalism, socialism and the modern welfare state

The desire to harm, a motivation for conduct

The teaching is:

On modern intellectualism

On Homosexuality

On Self-sufficient Country Living, Homesteading

Principles for Living Life

Topically Arranged Proverbs, Precepts, Quotations. Common Sayings. Poor Richard's Almanac.

America has lost her way

The really big sins

Theory on the Formation of Character

Moral Perversion

You are what you eat

People are like radio tuners --- they pick out and listen to one wavelength and ignore the rest

Cause of Character Traits --- According to Aristotle

These things go together

Television

We are what we eat --- living under the discipline of a diet

Avoiding problems and trouble in life

Role of habit in formation of character

The True Christian

What is true Christianity?

Personal attributes of the true Christian

What determines a person's character?

Love of God and love of virtue are closely united

Walking a solitary road

Intellectual disparities among people and the power in good habits

Tools of Satan. Tactics and Tricks used by the Devil.

On responding to wrongs

Real Christian Faith

The Natural Way -- The Unnatural Way

Wisdom, Reason and Virtue are closely related

Knowledge is one thing, wisdom is another

My views on Christianity in America

The most important thing in life is understanding

Sizing up people

We are all examples --- for good or for bad

Television --- spiritual poison

The Prime Mover that decides "What We Are"

Where do our outlooks, attitudes and values come from?

Sin is serious business. The punishment for it is real. Hell is real.

Self-imposed discipline and regimentation

Achieving happiness in life --- a matter of the right strategies

Self-discipline

Self-control, self-restraint, self-discipline basic to so much in life

We are our habits

What creates moral character?


[ Home ] [ Up ] [ Info ] [ Mail ]