# Linear Functional

(redirected from Linear functionals)

## 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.
The first step of enriching the class of uniform measures on convex domains to that of non-negatively curved (""log-concave"") measures in Euclidean space has been very successfully implemented in the last decades, leading to substantial progress in our understanding of volumetric properties of convex domains, mostly regarding concentration of linear functionals.
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.
Other topics include superselection rules induced by infrared divergence, Hermitian modifications of Toeplitz linear functionals and orthogonal polynomials, Dirac equations in cosmological backgrounds, and non-orthogonal signal representation.
We introduce fuzzy linear transformations, fuzzy linear functionals and fuzzy linear operators based on the notions of fuzzy fields and fuzzy linear spaces introduced by Gu Wenxiang and Lu Tu [1].
We consider linear functionals such that the corresponding measures are supported on an arc of the unit circle which doesn't contain [b.
In a manner similar to that in [P04a, PV05], we view this bijection as a map between integer points in polytopes which preserves certain linear functionals.
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.

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