Linear Functional

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linear functional

[′lin·ē·ər ′fəŋk·shən·əl]
(mathematics)
A linear transformation from a vector space to its scalar field.

Linear Functional

 

a generalization of the concept of linear form to vector spaces. A number-valued function f defined on a normed vector space E is called a linear functional on E if

(1) f(x) is linear, that is,

f(αx + βγ) = αf(x) + βf(y)

where x and y are any element of E and a and β are numbers, and

(2) f(x) is continuous. The continuity of f is equivalent to the requirement that ǀf(x)ǀ/ǀǀxǀǀ be bounded on E; in the latter case, the quantity

is called the norm of f and designated by ǀǀfǀǀ. Let C[a, b] be the space of the functions α(t), continuous for atb, with norm

Then the expressions

yield examples of linear functionals. In Hilbert space H the class of linear functionals coincides with the class of scalar products (l, x), where l is any fixed element of H.

In many problems it follows from general considerations that a certain quantity defines a linear functional. For example, solution of linear differential equations with linear boundary conditions leads to linear functionals. Therefore, the question of a general analytic expression for a linear functional in various spaces is of great significance.

The set of linear functionals on a given space E is made into a normed vector space E by introducing natural definitions of addition of linear functionals and their multiplication by numbers. The space E is called the adjoint of E; this space plays a major role in the study of E.

The concept of weak convergence involves linear functionals. Thus, a sequence {xn} of elements of a normed vector space is said to be weakly convergent to the element x if

for any linear functional f.

References in periodicals archive ?
It describes the linear or vector space concepts of addition and scalar multiplication, linear subspaces, linear functionals, and hyperplanes, as well as different distances in n-space and the geometric properties of subsets, subspaces, and hyperplanes; topology in the context of metrics derived from a norm on the n-dimensional space; the concept of convexity and the basic properties of convex sets; and Helly's theorem and applications involving transversals of families of pairwise disjoint compact convex subsets of the plane.
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O'Regan, Fixed point theory for set valued mappings between topological vector spaces having sufficiently many linear functionals, Computers and Mathematics with Applications 41 (2001), 917-928.
n] is a unitary operator F such that the linear functionals [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] do not define bounded linear functionals.
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For any linear functional u and any polynomial h, let Du = u' and hu be the linear functionals defined by duality
With an abuse of notations, we may introduce the linear functionals [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] associated with the measures [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] respectively, as follows:
Since [Mathematical Expression Omitted] is a finite family of continuous linear functionals on X and J is contained in the closure of I, it follows for p in J that there exists q in I with
This book is based on the author's own class-tested material and uses clear language to explain the major concepts of functional analysis, including Banach spaces, Hilbert spaces, topological vector spaces, as well as bounded linear functionals and operators.