Dual spaces

Definition 1. A linear functional on a vector space is a scalar-valued function defined for every vector , with the property that (identically in the vectors and and the scalars and )

Let us look at some examples of linear functionals.

Example 1. For in , write . More generally, let be any scalars and write

We observe that for any linear functional on any vector space for this reason a linear functional, as we defined it, is sometimes called homogeneous . In particular in , if is defined by then is not a linear functional unless .

Example 2. For any polynomial in , write . More generally, let be any scalars, let be any real numbers, and write

Another example, in a sense a limiting case of the one just given, is obtained as follows. Let be any finite interval on the real -axis, and let be any complex-valued integrable function defined on ; define by

Example 3. On an arbitrary vector space , define by writing for every in .

The last example is the first hint of a general situation. Let be any vector space and let be the collection of all linear functionals on . Let us denote by the linear functional defined in (3) (compare the comment at the end of Section: Comments ). If and are linear functionals on and if and are scalars, let us write for the function defined by

It is easy to see that is a linear functional; we denote it by . With these definitions of the linear concepts (zero, addition, scalar multiplication), the set forms a vector space, the dual space of .

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