Dual spaces

Definition 1. A linear functional on a vector space $𝒱$ 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 $α_{1}$ and $α_{2}$ ) $y (α_{1} x_{1} + α_{2} x_{2}) = α_{1} y (x_{1}) + α_{2} y (x_{2})$

Let us look at some examples of linear functionals.

Example 1. For $x = (ξ_{1}, \dots, ξ_{n})$ in $ℂ^{n}$ , write $y (x) = ξ_{1}$ . More generally, let $α_{1}, \dots, α_{n}$ be any $n$ scalars and write $y (x) = α_{1} ξ_{1} + \dots + α_{n} ξ_{n} .$

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 $ℂ^{n}$ , if $y$ is defined by $y (x) = α_{1} ξ_{1} + \dots + α_{n} ξ_{n} + β$ then $y$ is not a linear functional unless $β = 0$ .

Example 2. For any polynomial $x$ in $𝒫$ , write $y (x) = x (0)$ . More generally, let $α_{1}, \dots, α_{n}$ be any $n$ scalars, let $t_{1}, \dots, t_{n}$ be any $n$ real numbers, and write $y (x) = α_{1} x (t_{1}) + \dots + α_{n} x (t_{n}) .$

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 $α$ be any complex-valued integrable function defined on $(a, b)$ ; define $y$ by $y (x) = \int_{a}^{b} α (t) x (t) d t$

Example 3. On an arbitrary vector space $𝒱$ , define $y$ by writing $y (x) = 0$ for every $x$ in $𝒱$ .

The last example is the first hint of a general situation. Let $𝒱$ be any vector space and let \mathcal{V}^{\prime} be the collection of all linear functionals on $𝒱$ . 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 $𝒱$ and if $α_{1}$ and $α_{2}$ are scalars, let us write $y$ for the function defined by

$y (x) = α_{1} y_{1} (x) + α_{2} y_{2} (x) .$

It is easy to see that $y$ is a linear functional; we denote it by $α_{1} y_{1} + α_{2} y_{2}$ . With these definitions of the linear concepts (zero, addition, scalar multiplication), the set \mathcal{V}^{\prime} forms a vector space, the dual space of $𝒱$ .

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