Linear Algebra, 5th Edition

2.6* Dual Spaces

In this section, we are concerned exclusively with linear transformations from a vector space V into its field of scalars F, which is itself a vector space of dimension 1 over F. Such a linear transformation is called a linear functional on V. We generally use the letters f, g, h, ... to denote linear functionals. As we see in Example 1, the definite integral provides us with one of the most important examples of a linear functional in mathematics.

Example 1

Let V be the vector space of continuous real-valued functions on the interval $[0,2\pi ]$. Fix a function $g\in \text{V}$. The function $\text{h}:\text{V}\to R$ defined by

$$\text{h}(x)={\displaystyle \frac{1}{2\pi }}{\displaystyle {\int }_0^{2\pi }x(t)g(t)dt}$$

is a linear functional on V. In the cases that g(t) equals sin nt or cos nt, h(x) is often called the nth Fourier ...