Website owner:  James Miller

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

Functional. Linear functional. Continuous linear functional. Conjugate space, dual space. Dual basis. Annihilator.

Def. Functional. A functional is a function T:A → F in which the domain A consists of a set of functions and the set F is a number field. In other words, it is a functional assignment that assigns to every function of some set a number.


1. A function assigning to every function y = f(x), (a ole.gif x ole1.gif b) the arc length of the curve it describes

2. A function assigning to every function y = f(x), (a ole2.gif x ole3.gif b) the value of its definite integral


3. In Hilbert space the inner product


where g(x) is a given, fixed function.

4. The problems of the calculus of variations in which we seek maxima and minima of sets of functions.

If we view a function f(x) as a point in an infinite dimensional space, then a functional is simply a function of the points of an infinite-dimensional space. From this point of view the problems of the calculus of variations are concerned with the search for maxima and minima of functions of points of an infinite-dimensional space.

Note. Different authors define the term functional differently and there is substantial variation.

Def. Linear functional. A functional T:V → F defined on a vector space V is linear if

            T(av1 + bv2) = aTv1 + bTv2

for all vectors v1 and v2 and scalars a, b.

Def. Continuous linear functional. Norm of a functional. If a linear functional f :V → F has real or complex number values, then f is continuous for each x ε V if and only if there exists a number M such that


for each x (where ||x|| is the norm of x). The least such number M is called the norm of f and written || f ||.


Conjugate space. Let f be a continuous linear functional defined on a normed linear space N. The set of all such functionals is a complete normed linear space , or a Banach space, and is called the first conjugate space of N. The first conjugate space of this space is the second conjugate space of N, etc.

Syn. Adjoint space, dual space

If N is finite-dimensional, then N and its second conjugate space are identical (i.e. isometric).

For any normed linear space N, N is isometric with a subspace of its second conjugate space.

If N is a Hilbert space with a complete orthonormal sequence u1, u2, ..... , then the sequence of functions

            fn(x) = (x, un) [i.e. inner product x• un],      n = 1, 2, .....

is a complete orthonormal sequence in the first conjugate space and the correspondence


is an isometric correspondence between the two spaces.

                                                                        James & James. Mathematics Dictionary

Dual basis. (1) For a finite-dimensional linear space V with a basis (x1, x2, ..... , xn ), the dual basis is the set of linear functionals {f1, f2, ..... , fn }defined by


The dual basis is a basis for the first conjugate space V*. See Basis for Dual Space.

(2) If a Banach space (i.e. complete normed linear space ) has a basis (x1, x2, ..... ), then the sequence {f1, f2, ..... }defined by


is a sequence of continuous linear functionals and it is a basis (a dual basis) for the first conjugate space if and only if it is shrinking in the sense that


for each continuous functional f, where || f || n is the norm of f as a continuous linear functional with domain the linear span of {xn+1, xn+2, ....}. This condition is satisfied by all bases in reflective spaces (Hilbert space is a reflective Banach space). If {xα }is a complete orthonormal set for an inner product space T, then {fα }is a complete orthonormal set for the first conjugate space of T, where


Analogously to (1), each of the orthonormal bases {xα }and {fα } is dual to the other.

                                                                                    James & James. Mathematics Dictionary

Annihilator. An annihilator of a set S is the class of all functions of a certain type which annihilate S in the sense of being zero at each point of S. For example, if S is a subset of a normed linear space N, then the annihilator of S is the linear subset S' of the first conjugate space N* consisting of all continuous linear functionals which are zero at each point of S.


Let W be a subset (not necessarily a subspace) of a vector space V and V* be the first conjugate space of V. A linear functional f ole12.gif V* is called an annihilator of W if f(w) = 0 for every w ole13.gif W. The set of all such mappings, denoted by W0 and called the annihilator of W, is a subspace of V*.

Example 1. Pass a line through the origin of an x-y-z Cartesian coordinate system and label it L. Let W be the set of all vectors in line L. Pass a plane through the origin of the coordinate system perpendicular to line L and label it K. Let S represent the set of all vectors in plane K. Then S is an annihilator of W. Why? If s is any vector in S and w is any vector in W then the dot product s∙w = 0. The vector s can be viewed as a linear operator (linear functional) mapping the vectors of W into the field of reals and it maps all the elements of W into zero. By the same logic W is an annihilator of S. So the annihilator W0 of W is the set S and the annihilator S0 of the set S is the set W.

Example 2. The annihilator of a linear subset S of Hilbert space is the orthogonal complement of S.


  James & James. Mathematics Dictionary

More from

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


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

We are our habits

What creates moral character?

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