# Linear Functional

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## linear functional

[′lin·ē·ər ′fəŋk·shən·əl]## 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 ǀ*f*(*x*)ǀ/ǀǀ*x*ǀǀ be bounded on *E*; in the latter case, the quantity

is called the norm of *f* and designated by ǀǀ*f*ǀǀ. Let *C*[*a, b*] be the space of the functions *α*(*t*), continuous for *a* ≤ *t* ≤ *b*, 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 {*x _{n}*} of elements of a normed vector space is said to be weakly convergent to the element

*x*if

for any linear functional *f*.