Some Properties of Linear Functionals

Let us consider a linear space A = x of elements, e.g. a finite space or a Banach space with a basis. Any mapping x — 1 of the objects x on the field of complex numbers is referred to as afunctional, and such a mapping l(x) is called a linear functional if it satisfies the relation 

At this point it is convenient to introduce a new notation. If l(x) is the value of the linear functional in the point x, then l(x) = [11 x] may be considered as the dual product of the elements 1 and x, and one has immediately the two theorems  

It is evident that also Td is a linear operator, and that one further has the relations 

In the following we will study these concepts in greater detail. 

Vector representations. - Let us now assume that the original space A = x has a basis X = Xi, X2, X3,. .., so that one has an expansion theorem of the form 

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Some Properties of Linear Functionals and Adjoint Operators. 

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