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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.

Examples.

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

2. A function assigning to every function y = f(x), (a x 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.

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 V* is called an annihilator of W if f(w) = 0 for every w 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.

References

James & James. Mathematics Dictionary