Linear Functional

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linear functional

[′lin·ē·ər ′fəŋk·shən·əl]
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 can be used for a one-semester course at the graduate or senior undergraduate level in several topics, among them linear functional analysis, linear and nonlinear boundary value problems, and differential calculus and applications.
alpha]], via a suitable linear functional of umbral type [14], yields to a parametrized extension of classical orthogonal polynomials.
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n)] result from the positivity of a linear functional of such OPs.
of Barcelona) sets out the basic facts of linear functional analysis and its applications to some fundamental aspects of mathematical analysis, for graduate students of mathematics familiar with general topology, integral calculus with Lebesgue measure, and elementary aspects of normed or Hilbert spaces.
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v] for all y [member of] Y, then (T, (V, Y), (F X), (F, X)) is a fuzzy linear functional.
is non-trivial continuous linear functional for every y [member of] Y.
converges to the linear functional [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], i.
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n]] is the concentration of linear functional compounds and [[Mu].