Definition 1. A linear functional on a vector space \(\mathcal{V}\) is a scalar-valued function \(y\) defined for every vector \(x\) , with the property that (identically in the vectors \(x_{1}\) and \(x_{2}\) and the scalars \(\alpha_{1}\) and \(\alpha_{2}\) ) \[y(\alpha_{1} x_{1}+\alpha_{2} x_{2})=\alpha_{1} y(x_{1})+\alpha_{2} y(x_{2})\]
Let us look at some examples of linear functionals.
Example 1. For \(x=(\xi_{1}, \ldots, \xi_{n})\) in \(\mathbb{C}^{n}\) , write \(y(x)=\xi_{1}\) . More generally, let \(\alpha_{1}, \ldots, \alpha_{n}\) be any \(n\) scalars and write \[y(x)=\alpha_{1} \xi_{1}+\cdots+\alpha_{n} \xi_{n}.\]
We observe that for any linear functional \(y\) on any vector space \[y(0)=y(0\cdot 0)=0 \cdot y(0)=0;\] for this reason a linear functional, as we defined it, is sometimes called homogeneous . In particular in \(\mathbb{C}^{n}\) , if \(y\) is defined by \[y(x)=\alpha_{1} \xi_{1}+\cdots+\alpha_{n} \xi_{n}+\beta\] then \(y\) is not a linear functional unless \(\beta=0\) .
Example 2. For any polynomial \(x\) in \(\mathcal{P}\) , write \(y(x)=x(0)\) . More generally, let \(\alpha_{1}, \ldots, \alpha_{n}\) be any \(n\) scalars, let \(t_{1}, \ldots, t_{n}\) be any \(n\) real numbers, and write \[y(x)=\alpha_{1} x(t_{1})+\cdots+\alpha_{n} x(t_{n}).\]
Another example, in a sense a limiting case of the one just given, is obtained as follows. Let \((a, b)\) be any finite interval on the real \(t\) -axis, and let \(\alpha\) be any complex-valued integrable function defined on \((a, b)\) ; define \(y\) by \[y(x)=\int_{a}^{b} \alpha(t) x(t)\, dt\]
Example 3. On an arbitrary vector space \(\mathcal{V}\) , define \(y\) by writing \[y(x)=0\] for every \(x\) in \(\mathcal{V}\) .
The last example is the first hint of a general situation. Let \(\mathcal{V}\) be any vector space and let \(\mathcal{V}^{\prime}\) be the collection of all linear functionals on \(\mathcal{V}\) . Let us denote by \(0\) the linear functional defined in (3) (compare the comment at the end of Section: Comments ). If \(y_{1}\) and \(y_{2}\) are linear functionals on \(\mathcal{V}\) and if \(\alpha_{1}\) and \(\alpha_{2}\) are scalars, let us write \(y\) for the function defined by
\[y(x)=\alpha_{1} y_{1}(x)+\alpha_{2} y_{2}(x).\]
It is easy to see that \(y\) is a linear functional; we denote it by \(\alpha_{1} y_{1}+\alpha_{2} y_{2}\) . With these definitions of the linear concepts (zero, addition, scalar multiplication), the set \(\mathcal{V}^{\prime}\) forms a vector space, the dual space of \(\mathcal{V}\) .