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 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.