Brackets

Before studying linear functionals and dual spaces in more detail, we wish to introduce a notation that may appear weird at first sight but that will clarify many situations later on. Usually we denote a linear functional by a single letter such as . Sometimes, however, it is necessary to use the function notation fully and to indicate somehow that if is a linear functional on and if is a vector in , then is a particular scalar. According to the notation we propose to adopt here, we shall not write followed by in parentheses, but, instead, we shall write and enclosed between square brackets and separated by a comma. Because of the unusual nature of this notation, we shall expend on it some further verbiage.

As we have just pointed out is a substitute for the ordinary function symbol ; both these symbols denote the scalar we obtain if we take the value of the linear function at the vector . Let us take an analogous situation (concerned with functions that are, however, not linear). Let be the real function of a real variable defined for each real number by . The notation is a symbolic way of writing down the recipe for actual operations performed; it corresponds to the sentence [take a number, and square it].

Using this notation, we may sum up: to every vector space we make correspond the dual space consisting of all linear functionals on ; to every pair, and , where is a vector in and is a linear functional in , we make correspond the scalar defined to be the value of at . In terms of the symbol the defining property of a linear functional is

and the definition of the linear operations for linear functionals is

The two relations together are expressed by saying that is a bilinear functional of the vectors in and in .

EXERCISES

Exercise 1. Consider the set of complex numbers as a real vector space (as in Section: Examples , (9)). Suppose that for each in (where and are real numbers and ) the function is defined by

,
,
,
. (The square root sign attached to a positive number always denotes the positive square root of that number.)

In which of these cases is a linear functional?

Exercise 2. Suppose that for each in the function is defined by

In which of these cases is a linear functional?

Exercise 3. Suppose that for each in the function is defined by

In which of these cases is a linear functional?

Exercise 4. If is an arbitrary sequence of complex numbers, and if is an element of , , write . Prove that is an element of and that every element of can be obtained in this manner by a suitable choice of the ’s.

Exercise 5. If is a non-zero linear functional on a vector space , and if is an arbitrary scalar, does there necessarily exist a vector in such that ?

Exercise 6. Prove that if and are linear functionals (on the same vector space) such that whenever , then there exists a scalar such that . (Hint: if , write .)

EXERCISES

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